25. ∞ (−1)" (n+1) 3″ 22n+1 n=1 Answer 26. ☎ (-1)" √ñ In n n=2 27. 27, 28, 29, 30, and 31 Find the sum of the series. (-3) ¹-1 23n n=1
25. ∞ (−1)" (n+1) 3″ 22n+1 n=1 Answer 26. ☎ (-1)" √ñ In n n=2 27. 27, 28, 29, 30, and 31 Find the sum of the series. (-3) ¹-1 23n n=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve questions 25 and 27 for calculus. Image is attached.
![### Series Convergence Problems
**Instructions:** For each series below, determine whether it is absolutely convergent, conditionally convergent, or divergent.
**23.**
\[
\sum_{n=1}^{\infty} (-1)^{n-1}n^{-1/3}
\]
<br>
**24.**
\[
\sum_{n=1}^{\infty} (-1)^{n-1}n^{-3}
\]
<br>
**25.**
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1)3^n}{2^{2n+1}}
\]
<br>
**26.**
\[
\sum_{n=2}^{\infty} \frac{(-1)^{n}\sqrt{n}}{\ln n}
\]
---
### Series Sum Problems
**27.**
Find the sum of the series:
\[
\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{2^n}
\]
---
**Note:** Ensure to justify your answers using appropriate theorems and tests related to series convergence or divergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7ddc10c-4670-40fd-b02c-6a60c5fcc2f2%2F39c75342-ec0b-4e14-ac90-d8071b7a4f9c%2Fpa5p94_processed.png&w=3840&q=75)
Transcribed Image Text:### Series Convergence Problems
**Instructions:** For each series below, determine whether it is absolutely convergent, conditionally convergent, or divergent.
**23.**
\[
\sum_{n=1}^{\infty} (-1)^{n-1}n^{-1/3}
\]
<br>
**24.**
\[
\sum_{n=1}^{\infty} (-1)^{n-1}n^{-3}
\]
<br>
**25.**
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1)3^n}{2^{2n+1}}
\]
<br>
**26.**
\[
\sum_{n=2}^{\infty} \frac{(-1)^{n}\sqrt{n}}{\ln n}
\]
---
### Series Sum Problems
**27.**
Find the sum of the series:
\[
\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{2^n}
\]
---
**Note:** Ensure to justify your answers using appropriate theorems and tests related to series convergence or divergence.
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