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 series
a) \(\sum_{n=0}^{\infty}\left(\frac{4 n-1}{5 n+7}\right)^{n}\)
b) \(\sum_{n=0}^{\infty} \frac{\pi^{n}}{n !}\)
7. Use the Alternating Series Test to determine the convergence or divergence of the series
1. Determine whether the series converges or diverges.$$ \sum_{k=1}^{\infty} \frac{\ln (k)}{k} $$convergesdiverges2.Test the series for convergence or divergence.$$ \sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{3 \pi}{n}\right) $$convergesdiverges
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) \(\sum_{n=1}^{\infty} n \sin \frac{1}{n}\)
Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. 00 (-1)"n Σ n²-8 n=3 Identify an Evaluate the following limit. lima n- Since lim 0 and an - 12va, for all n. ---Select- Submit Answer
Use the Alternating Series Test, if applicable to determine the convergence or divergence of the series. (-1)"\n(n) n n2 Identify a Evaluate the following limit. lima Since lima 7.0 and an + 1 2 an for all n-Select-
11.) Use the Ratio Test to determine the convergence or divergence of the series (3n)! n=0 12.) Use the Root Test to determine the convergence or divergence of the series Š n =1
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) \(\sum_{n=1}^{\infty}\left[\cos \frac{1}{n^{2}}-\cos \frac{1}{(n+1)^{2}}\right]\)
9. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)a) \(\sum_{n=0}^{\infty}\left(\frac{3 x}{5}\right)^{n}\)b) \(\sum_{n=0}^{\infty} \frac{2^{n}(x-2)^{n}}{3 n}\)
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) \(\sum_{n=1}^{x} \frac{(2 n) !}{5^{n} n ! n t}\)
Use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test is inconclusive, determine the convergence or divergence of the series using other methods. (If you need to use oo or -oo, enter INFINITY or -INFINITY, respectively.) 0 5 n gh n = 1 a en + 1 lim n-> 00 a n
Use the Ratio Test to determine the convergence or divergence of the series. If the ratio Test is inconclusive, determine the convergence or divergence of the series using other methods. (If you need to use of c onter INFINITY - FINITY respectively) (n-1) verges dvoje Need Help? - 2 PUNIS LARCALLII 9.6.019.MI. MY NOTES ASK YOUR TEACHER Use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Tests inconclusive, determine the convergence or divergence...