1.
Determine whether the series converges or diverges.
$$ \sum_{k=1}^{\infty} \frac{\ln (k)}{k} $$
converges
diverges
2.
Test the series for convergence or divergence.
$$ \sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{3 \pi}{n}\right) $$
converges
diverges
Determine whether the given series converges or diverges. Fully justify your answe(a) \(\sum_{n=2}^{\infty} \frac{1}{\sqrt{n} \ln n}\)(b) \(\sum_{n=1}^{\infty} \cos \left(\frac{1}{n^{2}}\right)\)(c) \(\sum_{n=1}^{x} \frac{(2 n) !}{5^{n} n ! n t}\)
Determine whether the series converges or diverges.(1) \(\sum_{n=1}^{\infty} \frac{e^{1 / n}}{n^{2}}\)(2) \(\sum_{n=1}^{\infty}\left(\frac{2}{\sqrt{n}}+\frac{(-1)^{n}}{3^{n+1}}\right)\)(3) \(\sum_{n=1}^{\infty} \frac{5-2 \sin n}{n}\)(4) \(\sum_{n=1}^{\infty} \frac{3+\cos n}{n^{3 / 2}}\)(5) \(\sum_{n=0}^{\infty} \frac{\sqrt{n^{2}+2}}{n^{4}+n^{2}+5}\)(6) \(\sum_{n=1}^{\infty=1}\left(1+\frac{1}{n}\right)^{n}\)(7) \(\sum_{n=1}^{\infty} \frac{n+1}{n 2^{n}}\)(8) \(\sum_{n=1}^{\infty} \frac{\arctan n}{n^{4}}\)(9) \(\sum_{n=1}^{\infty} n \sin \frac{1}{n}\)
Determine whether the series converges, and if so, find its sum. (1) \(\sum_{n=1}^{\infty} 3^{-n} 8^{n+1}\)\((2) \sum_{n=2}^{\infty} \frac{1}{n(n-1)}\)(3) \(\sum_{n=0}^{\infty}(-3)\left(\frac{2}{3}\right)^{2 n}\)(4) \(\sum_{n=1}^{\infty} \frac{1}{e^{2 n}}\)(5) \(\sum_{n=1}^{\infty} \ln \frac{n}{n+1}\)(6) \(\sum_{n=1}^{\infty}[\arctan (n+1)-\arctan n]\)(7) \(\sum_{n=1}^{\infty} \ln \left(\frac{n^{2}+4}{2 n^{2}+1}\right)\)(8) \(\sum_{n=1}^{\infty} \frac{1+2^{n}}{3^{n}}\)(9) \(\sum_{n=1}^{\infty}\left[\cos \frac{1}{n^{2}}-\cos \frac{1}{(n+1)^{2}}\right]\)
7. Use the Alternating Series Test to determine the convergence or divergence of the series a) \(\sum_{n=1}^{\infty} \frac{(-1)^{n} \sqrt{n}}{2 n+1}\)b) \(\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{2 n-1}\)8. Use the Ratio Test or the Root Test to determine the convergence or divergence of the seriesa) \(\sum_{n=0}^{\infty}\left(\frac{4 n-1}{5 n+7}\right)^{n}\)b) \(\sum_{n=0}^{\infty} \frac{\pi^{n}}{n !}\)
Question 21 Indicate whether the series, \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n+ 3} converges or diverges. Select one: a. Converges b. Diverges
Use the Limit Comparison Test to determine whether the series converges or diverges. ∞ n = 1( n^0.6/ln(n))^ 2 Identify bn in the following limit n→∞ an/bn =? It's convergence or divergence?? We were unable to transcribe this imageWe were unable to transcribe this image
Determine whether the series is convergent or divergent.$$ \sum_{n=1}^{\infty}\left(\frac{8}{e^{n}}+\frac{4}{n(n+1)}\right) $$convergentdivergentIf it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)
Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) $$ \begin{gathered} a_{n}=\ln \left(2 n^{2}+6\right)-\ln \left(n^{2}+6\right) \\ \lim _{n \rightarrow \infty} a_{n}= \end{gathered} $$
k sin2 k diverges k1 1+k3 1. Determine whether the series converges or divergence -1(-1)"(Vn+1-Vm). 2. Test the series for convergence or k sin2 k diverges k1 1+k3 1. Determine whether the series converges or divergence -1(-1)"(Vn+1-Vm). 2. Test the series for convergence or
If the series \(\sum_{n=1}^{\infty} a_{n}\) converges and \(a_{n}>0\) for all \(n\), which of the following must be true?(A) \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=0\)(B) \(\left|a_{n}\right|<1\)for all \(n\)(C) \(\sum_{n=1}^{\infty} a_{n}=0\)(D) \(\sum_{n=1}^{\infty} n a_{n}\) diverges.(E) \(\sum_{n=1}^{\infty} \frac{a_{n}}{n}\) converges.