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What are convergent series and what are absolutely convergent series?
A convergent series is a series of numbers that has a finite sum. In other words, as you add up more and more terms of the series, the sum approaches a specific value. On the other hand, an absolutely convergent series is a series in which the absolute values of the terms converge to a finite sum. In other words, the series converges when you take the absolute value of each term and then add them up. Absolutely convergent series have the property that rearranging the terms does not change the sum, while for convergent series, rearranging the terms can change the sum. **
Is the product of two convergent sequences always a convergent sequence?
No, the product of two convergent sequences is not always a convergent sequence. While the product of two convergent sequences may converge, it is not guaranteed. This is because the convergence of a product of sequences depends on the behavior of the individual sequences and their interaction with each other. Therefore, it is possible for the product of two convergent sequences to be divergent. **
Similar search terms for Convergent
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Apple AirPods Max 2 Wireless Over-Ear Headphones Starlight – Active Noise Cancellation, Adaptive Audio, Live Translation & Personalised Spatial Audio.gadcet-pdp{--g-green:#00a651;color:#1f2430;line-height:1.55;font-size:inherit;max-width:760px}.gadcet-pdp>*:first-child{margin-top:0}.gadcet-pdp p{margin:0 0.9em}.gadcet-pdp h3{font-size:1.05em;font-weight:600;color:#11161f;margin:1.4em 0.5em;padding-left:.5em;border-left:3px solid var(--g-green);border-radius:0;line-height:1.3}.gadcet-pdp ul{list-style:none!important;margin:0 0.9em;padding:0!important}.gadcet-pdp ul li{position:relative;padding:.1em 0.1em 1.5em;margin:0}.gadcet-pdp ul li::before{content:"";position:absolute;left:.16em;top:.45em;width:.36em;height:.66em;border:solid var(--g-green);border-width:0.14em.14em 0;transform:rotate(45deg)}.gadcet-pdp table{width:100%;border-collapse:collapse;margin:.3em 0 1em;font-size:.95em;border:1px solid #e7e9ee!important;border-radius:8px;overflow:hidden}.gadcet-pdp table td{padding:.5em.8em;border-bottom:1px solid #eef0f4!important;vertical-align:top}.gadcet-pdp table tr:last-child td{border-bottom:0!important}.gadcet-pdp table tr:nth-child(even){background:#f7f9f8}.gadcet-pdp table td:first-child{font-weight:600;color:#454c59;width:40%}@media (max-width:600px){.gadcet-pdp table td:first-child{width:42%}} Experience immersive Apple audio with the AirPods Max 2 Wireless Over-Ear Headphones in Starlight — built for focused listening, clearer calls and seamless everyday use with Active Noise Cancellation, Adaptive Audio and Personalised Spatial Audio. Key Features Active Noise Cancellation helps reduce unwanted background noise for more focused listening. Adaptive Audio intelligently blends noise control and awareness as your surroundings change. Personalised Spatial Audio with dynamic head tracking creates a more immersive soundstage for supported content. Live Translation support for real-time conversations across selected languages when paired with a compatible Apple Intelligence-enabled iPhone. Apple H2 headphone chip in each ear cup for intelligent audio processing and smooth Apple device use. Bluetooth 5.3 wireless technology for cable-free listening. USB-C charging, plus USB-C wired listening for lossless audio and ultra-low latency with compatible content and apps. Up to 20 hours of listening time with Active Noise Cancellation enabled. A 5-minute charge provides around 1.5 hours of listening time. Nine microphones support noise control and voice pickup for calls, meetings and voice commands. Benefits Stay immersed on commutes, flights, study sessions and busy workdays. Enjoy films, music and supported content with a wider, more cinematic feel. Move smoothly between compatible Apple devices such as iPhone, iPad and Mac. Brand new, ready to enjoy straight from the box. Specifications Brand Apple Model AirPods Max 2 Colour Starlight Type Wireless over-ear headphones Audio Features Active Noise Cancellation, Adaptive Audio, Transparency mode, Personalised Spatial Audio Chip Apple H2 headphone chip in each ear cup Microphones Nine microphones total Connectivity Bluetooth 5.3 wireless technology...467,00 £*Shipping: 0,00 £Secure redirect to the provider
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Sonos Five HiFi Speaker with WiFi, 3.5 mm Line In, Support for Hi-Res Audio Streaming - Black, New.gadcet-pdp{--g-green:#00a651;color:#1f2430;line-height:1.55;font-size:inherit;max-width:760px}.gadcet-pdp>*:first-child{margin-top:0}.gadcet-pdp p{margin:0 0.9em}.gadcet-pdp h3{font-size:1.05em;font-weight:600;color:#11161f;margin:1.4em 0.5em;padding-left:.5em;border-left:3px solid var(--g-green);border-radius:0;line-height:1.3}.gadcet-pdp ul{list-style:none!important;margin:0 0.9em;padding:0!important}.gadcet-pdp ul li{position:relative;padding:.1em 0.1em 1.5em;margin:0}.gadcet-pdp ul li::before{content:"";position:absolute;left:.16em;top:.45em;width:.36em;height:.66em;border:solid var(--g-green);border-width:0.14em.14em 0;transform:rotate(45deg)}.gadcet-pdp table{width:100%;border-collapse:collapse;margin:.3em 0 1em;font-size:.95em;border:1px solid #e7e9ee!important;border-radius:8px;overflow:hidden}.gadcet-pdp table td{padding:.5em.8em;border-bottom:1px solid #eef0f4!important;vertical-align:top}.gadcet-pdp table tr:last-child td{border-bottom:0!important}.gadcet-pdp table tr:nth-child(even){background:#f7f9f8}.gadcet-pdp table td:first-child{font-weight:600;color:#454c59;width:40%}@media (max-width:600px){.gadcet-pdp table td:first-child{width:42%}} Bring room-filling, high-fidelity sound to your home with the Sonos Five in Black. Enjoy high-resolution lossless audio streaming over WiFi, flexible wired playback and convenient multi-room listening. Key Features Six Class-D digital amplifiers tuned to the speaker's acoustic architecture Three tweeters for detailed highs and a wide stereo soundstage Three midwoofers for balanced vocals, instruments and bass WiFi streaming with support for high-resolution lossless audio 3.5 mm line-in for compatible turntables, CD players, computers and other audio devices Apple AirPlay 2 support for streaming from compatible Apple devices Trueplay tuning using a supported iOS device Horizontal or vertical placement with orientation-aware processing Can be paired with another Sonos Five for a wider stereo soundstage Capacitive touch controls and a 10/100 Ethernet port Benefits Enjoy powerful, detailed music playback with clear vocals and deep bass Connect wireless and wired audio sources to one versatile speaker Expand your listening across compatible Sonos speakers in multiple rooms Fine-tune bass, treble and loudness through the Sonos app Specifications Brand Sonos Model Five Product type HiFi wireless speaker Colour Black Finish Matte Amplifiers 6 Class-D digital amplifiers Tweeters 3 Midwoofers 3 Wireless connectivity WiFi and Apple AirPlay 2 Wired connectivity 3.5 mm line-in and 10/100 Ethernet WiFi support 802.11a/b/g/n, 2.4 GHz or 5 GHz router support EAN 8717755777140 What You'll Receive One Sonos Five HiFi speaker in Black, supplied in your selected New or Used - Like New condition. Ideal For Ideal for music lovers seeking high-quality home audio, multi-room listening or a speaker with line-in connectivity for compatible turntables, CD players and computers.548,98 £*Shipping: 0,00 £Secure redirect to the provider
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Is the series convergent?
To determine if a series is convergent, we need to analyze the behavior of its terms as the number of terms approaches infinity. If the terms of the series approach a finite value as the number of terms increases, then the series is convergent. On the other hand, if the terms do not approach a finite value, the series is divergent. **
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Determine whether the following series are absolutely convergent, conditionally convergent, or divergent.
To determine whether a series is absolutely convergent, conditionally convergent, or divergent, we need to consider both the original series and the absolute value of the series. If the original series converges and the absolute value of the series also converges, then the series is absolutely convergent. If the original series converges but the absolute value of the series diverges, then the series is conditionally convergent. If the original series diverges, then the series is divergent. **
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Is the series a convergent if b is a convergent positive sequence?
Yes, if b is a convergent positive sequence, then the series Σb_n will also be convergent. This is because the convergence of the sequence b_n implies that the terms of the sequence approach a finite limit as n goes to infinity. As a result, the terms of the series Σb_n will also approach zero, and the series will converge. Therefore, the convergence of the sequence b_n guarantees the convergence of the series Σb_n. **
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Is every convergent sequence monotonic?
No, not every convergent sequence is monotonic. A convergent sequence is one that approaches a specific limit as the number of terms in the sequence increases. A monotonic sequence, on the other hand, is one that is either always increasing or always decreasing. While some convergent sequences may be monotonic, there are also convergent sequences that oscillate or have a mix of increasing and decreasing terms as they approach their limit. Therefore, not every convergent sequence is monotonic. **
Is the alternating sequence convergent?
No, the alternating sequence is not necessarily convergent. An alternating sequence is a sequence in which the terms alternate in sign. Whether or not the alternating sequence converges depends on the behavior of the terms in the sequence. If the terms in the sequence do not approach a specific value as n approaches infinity, then the alternating sequence is not convergent. **
What is the proof that a rearrangement of an absolutely convergent series is also convergent?
The proof that a rearrangement of an absolutely convergent series is also convergent lies in the fact that absolute convergence implies convergence. Since the series is absolutely convergent, we know that the sum of the absolute values of the terms converges. Therefore, no matter how we rearrange the terms, the rearranged series will still converge to the same sum as the original series. This is because the convergence of the rearranged series is guaranteed by the convergence of the absolute values of the terms. **
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Products related to Convergent:
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What are convergent series and what are absolutely convergent series?
A convergent series is a series of numbers that has a finite sum. In other words, as you add up more and more terms of the series, the sum approaches a specific value. On the other hand, an absolutely convergent series is a series in which the absolute values of the terms converge to a finite sum. In other words, the series converges when you take the absolute value of each term and then add them up. Absolutely convergent series have the property that rearranging the terms does not change the sum, while for convergent series, rearranging the terms can change the sum. **
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Is the product of two convergent sequences always a convergent sequence?
No, the product of two convergent sequences is not always a convergent sequence. While the product of two convergent sequences may converge, it is not guaranteed. This is because the convergence of a product of sequences depends on the behavior of the individual sequences and their interaction with each other. Therefore, it is possible for the product of two convergent sequences to be divergent. **
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Is the series convergent?
To determine if a series is convergent, we need to analyze the behavior of its terms as the number of terms approaches infinity. If the terms of the series approach a finite value as the number of terms increases, then the series is convergent. On the other hand, if the terms do not approach a finite value, the series is divergent. **
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Determine whether the following series are absolutely convergent, conditionally convergent, or divergent.
To determine whether a series is absolutely convergent, conditionally convergent, or divergent, we need to consider both the original series and the absolute value of the series. If the original series converges and the absolute value of the series also converges, then the series is absolutely convergent. If the original series converges but the absolute value of the series diverges, then the series is conditionally convergent. If the original series diverges, then the series is divergent. **
Similar search terms for Convergent
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Apple AirPods Max 2 Wireless Over-Ear Headphones Starlight – Active Noise Cancellation, Adaptive Audio, Live Translation & Personalised Spatial Audio.gadcet-pdp{--g-green:#00a651;color:#1f2430;line-height:1.55;font-size:inherit;max-width:760px}.gadcet-pdp>*:first-child{margin-top:0}.gadcet-pdp p{margin:0 0.9em}.gadcet-pdp h3{font-size:1.05em;font-weight:600;color:#11161f;margin:1.4em 0.5em;padding-left:.5em;border-left:3px solid var(--g-green);border-radius:0;line-height:1.3}.gadcet-pdp ul{list-style:none!important;margin:0 0.9em;padding:0!important}.gadcet-pdp ul li{position:relative;padding:.1em 0.1em 1.5em;margin:0}.gadcet-pdp ul li::before{content:"";position:absolute;left:.16em;top:.45em;width:.36em;height:.66em;border:solid var(--g-green);border-width:0.14em.14em 0;transform:rotate(45deg)}.gadcet-pdp table{width:100%;border-collapse:collapse;margin:.3em 0 1em;font-size:.95em;border:1px solid #e7e9ee!important;border-radius:8px;overflow:hidden}.gadcet-pdp table td{padding:.5em.8em;border-bottom:1px solid #eef0f4!important;vertical-align:top}.gadcet-pdp table tr:last-child td{border-bottom:0!important}.gadcet-pdp table tr:nth-child(even){background:#f7f9f8}.gadcet-pdp table td:first-child{font-weight:600;color:#454c59;width:40%}@media (max-width:600px){.gadcet-pdp table td:first-child{width:42%}} Experience immersive Apple audio with the AirPods Max 2 Wireless Over-Ear Headphones in Starlight — built for focused listening, clearer calls and seamless everyday use with Active Noise Cancellation, Adaptive Audio and Personalised Spatial Audio. Key Features Active Noise Cancellation helps reduce unwanted background noise for more focused listening. Adaptive Audio intelligently blends noise control and awareness as your surroundings change. Personalised Spatial Audio with dynamic head tracking creates a more immersive soundstage for supported content. Live Translation support for real-time conversations across selected languages when paired with a compatible Apple Intelligence-enabled iPhone. Apple H2 headphone chip in each ear cup for intelligent audio processing and smooth Apple device use. Bluetooth 5.3 wireless technology for cable-free listening. USB-C charging, plus USB-C wired listening for lossless audio and ultra-low latency with compatible content and apps. Up to 20 hours of listening time with Active Noise Cancellation enabled. A 5-minute charge provides around 1.5 hours of listening time. Nine microphones support noise control and voice pickup for calls, meetings and voice commands. Benefits Stay immersed on commutes, flights, study sessions and busy workdays. Enjoy films, music and supported content with a wider, more cinematic feel. Move smoothly between compatible Apple devices such as iPhone, iPad and Mac. Brand new, ready to enjoy straight from the box. Specifications Brand Apple Model AirPods Max 2 Colour Starlight Type Wireless over-ear headphones Audio Features Active Noise Cancellation, Adaptive Audio, Transparency mode, Personalised Spatial Audio Chip Apple H2 headphone chip in each ear cup Microphones Nine microphones total Connectivity Bluetooth 5.3 wireless technology...467,00 £*Shipping: 0,00 £Secure redirect to the provider
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Sonos Five HiFi Speaker with WiFi, 3.5 mm Line In, Support for Hi-Res Audio Streaming - Black, New.gadcet-pdp{--g-green:#00a651;color:#1f2430;line-height:1.55;font-size:inherit;max-width:760px}.gadcet-pdp>*:first-child{margin-top:0}.gadcet-pdp p{margin:0 0.9em}.gadcet-pdp h3{font-size:1.05em;font-weight:600;color:#11161f;margin:1.4em 0.5em;padding-left:.5em;border-left:3px solid var(--g-green);border-radius:0;line-height:1.3}.gadcet-pdp ul{list-style:none!important;margin:0 0.9em;padding:0!important}.gadcet-pdp ul li{position:relative;padding:.1em 0.1em 1.5em;margin:0}.gadcet-pdp ul li::before{content:"";position:absolute;left:.16em;top:.45em;width:.36em;height:.66em;border:solid var(--g-green);border-width:0.14em.14em 0;transform:rotate(45deg)}.gadcet-pdp table{width:100%;border-collapse:collapse;margin:.3em 0 1em;font-size:.95em;border:1px solid #e7e9ee!important;border-radius:8px;overflow:hidden}.gadcet-pdp table td{padding:.5em.8em;border-bottom:1px solid #eef0f4!important;vertical-align:top}.gadcet-pdp table tr:last-child td{border-bottom:0!important}.gadcet-pdp table tr:nth-child(even){background:#f7f9f8}.gadcet-pdp table td:first-child{font-weight:600;color:#454c59;width:40%}@media (max-width:600px){.gadcet-pdp table td:first-child{width:42%}} Bring room-filling, high-fidelity sound to your home with the Sonos Five in Black. Enjoy high-resolution lossless audio streaming over WiFi, flexible wired playback and convenient multi-room listening. Key Features Six Class-D digital amplifiers tuned to the speaker's acoustic architecture Three tweeters for detailed highs and a wide stereo soundstage Three midwoofers for balanced vocals, instruments and bass WiFi streaming with support for high-resolution lossless audio 3.5 mm line-in for compatible turntables, CD players, computers and other audio devices Apple AirPlay 2 support for streaming from compatible Apple devices Trueplay tuning using a supported iOS device Horizontal or vertical placement with orientation-aware processing Can be paired with another Sonos Five for a wider stereo soundstage Capacitive touch controls and a 10/100 Ethernet port Benefits Enjoy powerful, detailed music playback with clear vocals and deep bass Connect wireless and wired audio sources to one versatile speaker Expand your listening across compatible Sonos speakers in multiple rooms Fine-tune bass, treble and loudness through the Sonos app Specifications Brand Sonos Model Five Product type HiFi wireless speaker Colour Black Finish Matte Amplifiers 6 Class-D digital amplifiers Tweeters 3 Midwoofers 3 Wireless connectivity WiFi and Apple AirPlay 2 Wired connectivity 3.5 mm line-in and 10/100 Ethernet WiFi support 802.11a/b/g/n, 2.4 GHz or 5 GHz router support EAN 8717755777140 What You'll Receive One Sonos Five HiFi speaker in Black, supplied in your selected New or Used - Like New condition. Ideal For Ideal for music lovers seeking high-quality home audio, multi-room listening or a speaker with line-in connectivity for compatible turntables, CD players and computers.548,98 £*Shipping: 0,00 £Secure redirect to the provider
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Lisciani Barbie Bluetooth Headphones for Kids, Kid-Friendly Wireless Audio, Ages Four and UpLisciani Barbie Bluetooth headphones bring kid-friendly wireless audio to playtime for children aged four and up. The Lisciani brand combines a trusted toy heritage with practical tech, delivering Bluetooth headphones designed for kids and easy for parents to trust. These headphones are designed for everyday adventures and feature a durable, lightweight construction that fuses Barbie-themed style with comfort. They're perfect for car rides, classroom activities, or quiet-time listening. The headphones pair with a smartphone or tablet in seconds and switch devices with ease. A safe volume limiter protects young hearing during hours of listening. The headphones have a lightweight, adjustable headband with soft, cushioned ear cups for all-day comfort. The Barbie-themed design is a familiar look that kids will love to wear. They're ideal for kids aged four and up.24,49 £*Shipping: 3,95 £Secure redirect to the provider
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ECLAT Tech Ltd Apple AirPods Max 2 Wireless Over-Ear Headphones Starlight – Active Noise Cancellation, Adaptive Audio, Live Translation & Personalised Spatial Audio, New.gadcet-pdp{--g-green:#00a651;color:#1f2430;line-height:1.55;font-size:inherit;max-width:760px}.gadcet-pdp>*:first-child{margin-top:0}.gadcet-pdp p{margin:0 0.9em}.gadcet-pdp h3{font-size:1.05em;font-weight:600;color:#11161f;margin:1.4em 0.5em;padding-left:.5em;border-left:3px solid var(--g-green);border-radius:0;line-height:1.3}.gadcet-pdp ul{list-style:none!important;margin:0 0.9em;padding:0!important}.gadcet-pdp ul li{position:relative;padding:.1em 0.1em 1.5em;margin:0}.gadcet-pdp ul li::before{content:"";position:absolute;left:.16em;top:.45em;width:.36em;height:.66em;border:solid var(--g-green);border-width:0.14em.14em 0;transform:rotate(45deg)}.gadcet-pdp table{width:100%;border-collapse:collapse;margin:.3em 0 1em;font-size:.95em;border:1px solid #e7e9ee!important;border-radius:8px;overflow:hidden}.gadcet-pdp table td{padding:.5em.8em;border-bottom:1px solid #eef0f4!important;vertical-align:top}.gadcet-pdp table tr:last-child td{border-bottom:0!important}.gadcet-pdp table tr:nth-child(even){background:#f7f9f8}.gadcet-pdp table td:first-child{font-weight:600;color:#454c59;width:40%}@media (max-width:600px){.gadcet-pdp table td:first-child{width:42%}} Experience immersive Apple audio with the AirPods Max 2 Wireless Over-Ear Headphones in Starlight, built for focused listening, clearer calls and seamless everyday use. With Active Noise Cancellation, Adaptive Audio and Personalised Spatial Audio, they bring a premium feel to music, films, work and travel. Key Features Active Noise Cancellation helps reduce unwanted background noise for more focused listening. Adaptive Audio intelligently blends noise control and awareness as your surroundings change. Personalised Spatial Audio with dynamic head tracking creates a more immersive soundstage for supported content. Live Translation support helps with real-time conversations across selected languages when paired with a compatible Apple Intelligence-enabled iPhone. Apple H2 headphone chip in each ear cup supports intelligent audio processing and smooth Apple device use. Bluetooth 5.3 wireless technology for cable-free listening. USB-C charging, with USB-C wired listening support for lossless audio and ultra-low latency with compatible content and apps. Up to 20 hours listening time with Active Noise Cancellation enabled. A 5-minute charge provides around 1.5 hours of listening time. Nine microphones support noise control and voice pickup for calls, meetings and voice commands. Benefits Stay immersed on commutes, flights, study sessions and busy workdays. Enjoy films, music and supported content with a wider, more cinematic feel. Move more smoothly between compatible Apple devices such as iPhone, iPad and Mac. Choose the condition that suits you, with New and Used - Like New options available. Specifications Brand Apple Model AirPods Max 2 Colour Starlight Headphone Type Wireless over-ear headphones Audio Features Active Noise Cancellation, Adaptive Audio, Transparency mode, Personalised Spatial Audio...469,00 £*Shipping: 0,00 £Secure redirect to the provider
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Is the series a convergent if b is a convergent positive sequence?
Yes, if b is a convergent positive sequence, then the series Σb_n will also be convergent. This is because the convergence of the sequence b_n implies that the terms of the sequence approach a finite limit as n goes to infinity. As a result, the terms of the series Σb_n will also approach zero, and the series will converge. Therefore, the convergence of the sequence b_n guarantees the convergence of the series Σb_n. **
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Is every convergent sequence monotonic?
No, not every convergent sequence is monotonic. A convergent sequence is one that approaches a specific limit as the number of terms in the sequence increases. A monotonic sequence, on the other hand, is one that is either always increasing or always decreasing. While some convergent sequences may be monotonic, there are also convergent sequences that oscillate or have a mix of increasing and decreasing terms as they approach their limit. Therefore, not every convergent sequence is monotonic. **
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Is the alternating sequence convergent?
No, the alternating sequence is not necessarily convergent. An alternating sequence is a sequence in which the terms alternate in sign. Whether or not the alternating sequence converges depends on the behavior of the terms in the sequence. If the terms in the sequence do not approach a specific value as n approaches infinity, then the alternating sequence is not convergent. **
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What is the proof that a rearrangement of an absolutely convergent series is also convergent?
The proof that a rearrangement of an absolutely convergent series is also convergent lies in the fact that absolute convergence implies convergence. Since the series is absolutely convergent, we know that the sum of the absolute values of the terms converges. Therefore, no matter how we rearrange the terms, the rearranged series will still converge to the same sum as the original series. This is because the convergence of the rearranged series is guaranteed by the convergence of the absolute values of the terms. **
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