91Ó°ÊÓ

It is known that the equation \(\sum_{n=0}^{\infty} n x^{n}=x /(x-1)^{2}\) holds for \(-1

Use a Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty}\left(\sec \left(\frac{1}{\sqrt{n}}\right)-\cos \left(\frac{1}{\sqrt{n}}\right)\right) $$

Use the Integral Test to determine whether the given series converges. $$ \sum_{n=1}^{\infty} \frac{1}{1+n^{1 / 2}} $$

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

Use partial fractions to calculate the \(N^{\text {th }}\) partial sum \(S_{N}\) of the given series in closed form. Sum the series by finding \(\lim _{N \rightarrow \infty} S_{N}\). $$ \sum_{n=1}^{\infty} \frac{1}{(2 n+1)(2 n+3)} $$

We know that \(\sum_{n=0}^{\infty}(-1)^{n} t^{n}=1 /(1+t)\) for \(-1

Use a Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=2}^{\infty}\left(\frac{1+\ln \left(n^{2}\right)}{\ln \left(n^{4}\right)}\right)^{n} $$

Use partial fractions to calculate the \(N^{\text {th }}\) partial sum \(S_{N}\) of the given series in closed form. Sum the series by finding \(\lim _{N \rightarrow \infty} S_{N}\). $$ \sum_{n=1}^{\infty} \frac{2 n+1}{\left(n^{2}+n\right)^{2}} $$

Find the Maclaurin series of \(\cosh (x)\) and \(\sinh (x)\).

Use the Integral Test to determine whether the given series converges. $$ \sum_{n=1}^{\infty} \ln \left(\frac{n+3}{n}\right) $$