#40 ## Series and Convergence Problems### Practice ProblemsBelow are several exercises designed to test your understanding of series and their convergence. Work through each problem and determine whether the series converges or diverges.#### Series Exercises:**33.** \[\sum_{n=1}^{8} \frac{(-9)^n}{n10^{n+1}}\]**34.** \[\sum_{n=1}^{8} \frac{n5^{2n}}{10^{n+1}}\]**35.** \[\sum_{n=2}^{8} \left(\frac{n}{\ln n}\right)^n\]**36.** \[\sum_{n=1}^{8} \frac{\sin(n\pi/6)}{1 + n\sqrt{n}}\]**37.** \[\sum_{n=1}^{8} \frac{(-1)^n \arctan n}{n^2}\]**38.** \[\sum_{n=2}^{8} \frac{(-1)^n}{n \ln n}\]#### Recursive Series Problems:**39.** The terms of a series are defined recursively by the equation:\[a_1 = 2 \quad \text{and} \quad a_{n+1} = \frac{5n + 1}{4n + 3} a_n\]**Determine whether** \(\sum a_n\) **converges or diverges.****40.** A series \(\sum a_n\) is defined by the equations:\[a_1 = 1 \quad \text{and} \quad a_{n+1} = \frac{2 + \cos n}{\sqrt{n}} a_n\]**Determine whether** \(\sum a_n\) **converges or diverges.**#### Series of Positive Numbers:For problems 41 and 42, let \(\{b_n\}\) be a sequence of positive numbers that converge to \(\frac{1}{2}\). Determine whether the given series is absolutely convergent.**41.** \[\sum_{n=1}^{\infty} \frac{b_n^n \cos n\pi}{n

Name: #40 ## Series and Convergence Problems### Practice ProblemsBelow are s
Uploaded: 2024-03-05T11:52:21.000Z
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Description: #40 ## Series and Convergence Problems### Practice ProblemsBelow are several exercises designed to test your understanding of series and their convergence. Work through each problem and determine whether the series converges or diverges.#### Series E