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6. Use the definition of convergence to prove that each of the following sequences converges: (a) (b) 5+- n 1 [2 - 2n10⁰ n

6. Use the definition of convergence to prove that each of the following sequences converges: (a) (b) 5+- n 1 [2 - 2n10⁰ n

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

#6 please. Thanks

The image contains two mathematical sequences: (c) \(\{2^{-n}\}_{n=1}^{\infty}\) This represents an infinite sequence starting from \(n=1\). The general term of the sequence is \(2^{-n}\), which signifies \(1 / (2^n)\). As \(n\) increases, the terms of the sequence get smaller and approach zero. (d) \(\left\{\frac{3n}{2n+1}\right\}_{n=1}^{\infty}\) This sequence is also infinite, beginning at \(n=1\). The general term is \(\frac{3n}{2n + 1}\). As \(n\) becomes larger, the value of the term approaches \(\frac{3}{2}\), since the highest degree terms in the numerator and denominator (both \(n\)) dominate the expression.
Transcribed Image Text:The image contains two mathematical sequences: (c) \(\{2^{-n}\}_{n=1}^{\infty}\) This represents an infinite sequence starting from \(n=1\). The general term of the sequence is \(2^{-n}\), which signifies \(1 / (2^n)\). As \(n\) increases, the terms of the sequence get smaller and approach zero. (d) \(\left\{\frac{3n}{2n+1}\right\}_{n=1}^{\infty}\) This sequence is also infinite, beginning at \(n=1\). The general term is \(\frac{3n}{2n + 1}\). As \(n\) becomes larger, the value of the term approaches \(\frac{3}{2}\), since the highest degree terms in the numerator and denominator (both \(n\)) dominate the expression.
### Educational Content on Sequences and Convergence 5. **Example Request**: - Provide an example of a sequence that is bounded but not convergent. 6. **Convergence Proof Task**: - Utilize the definition of convergence to demonstrate that each of the following sequences converges: **(a)** Sequence: \( \left\{ 5 + \frac{1}{n} \right\}_{n=1}^{\infty} \) **(b)** Sequence: \( \left\{ \frac{2 - 2n}{n} \right\}_{n=1}^{\infty} \) In tasks (a) and (b), you are required to use the formal definition of convergence. For a sequence \(\{a_n\}\) to converge to a limit \(L\), for every \(\epsilon > 0\), there must exist an integer \(N\) such that for all \(n > N\), \(|a_n - L| < \epsilon\). Analyze each sequence, determine its limit, and verify convergence according to this definition.
Transcribed Image Text:### Educational Content on Sequences and Convergence 5. **Example Request**: - Provide an example of a sequence that is bounded but not convergent. 6. **Convergence Proof Task**: - Utilize the definition of convergence to demonstrate that each of the following sequences converges: **(a)** Sequence: \( \left\{ 5 + \frac{1}{n} \right\}_{n=1}^{\infty} \) **(b)** Sequence: \( \left\{ \frac{2 - 2n}{n} \right\}_{n=1}^{\infty} \) In tasks (a) and (b), you are required to use the formal definition of convergence. For a sequence \(\{a_n\}\) to converge to a limit \(L\), for every \(\epsilon > 0\), there must exist an integer \(N\) such that for all \(n > N\), \(|a_n - L| < \epsilon\). Analyze each sequence, determine its limit, and verify convergence according to this definition.
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Step 1: Introduction of the given problem

Using definition we have to show that the given sequence is convergent.

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