91Ó°ÊÓ

Use the Limit Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{n^{2}+2}{\left(n^{2}+1\right)^{2}} $$

Compute the Taylor polynomial \(T_{N}\) of the given function \(f\) with the given base point \(c\) and given order \(N\). \(f(x)=x^{2}+1+1 / x^{2} \quad N=5 \quad c=-1\)

Use the Integral Test to determine whether the given series converges or diverges. Before you apply the test, be sure that the hypotheses are satisfied. $$ \sum_{n=2}^{\infty} \frac{1}{n \ln (n)} $$

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty}\left(1-\frac{1}{n}\right)^{n^{2}}\)

Find the sum of the given series. $$ \sum_{n=4}^{\infty}(0.2)^{n} $$

Use the Integral Test to determine whether the given series converges or diverges. Before you apply the test, be sure that the hypotheses are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{(n+3)^{5 / 4}} $$

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=3}^{\infty}(-1)^{n} \frac{1}{n \sqrt{\ln (n)}} $$

Compute the Taylor polynomial \(T_{N}\) of the given function \(f\) with the given base point \(c\) and given order \(N\). \(f(x)=\ln (3 x+7) \quad N=4 \quad c=-2\)

Find the sum of the given series. $$ \sum_{n=-3}^{\infty}(1 / 5)^{n} $$

Use the Limit Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{\sqrt{n}+1}{2 n^{2}-n} $$