91Ó°ÊÓ

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{3^{n}+n}{\sqrt{n} 3^{n}+1} $$

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=5 \cos (2 x)-4 \sin (3 x) $$

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. $$ \sum_{n=1}^{\infty} e^{-n} $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{2^{n}+1}{2^{n}+n^{2}} $$

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{10^{n}}{n^{10}}\)

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=e^{1-x} $$

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. $$ \sum_{n=1}^{\infty} \frac{n}{n^{2}+1} $$

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

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n}\left(\frac{e n+1}{\pi n+1}\right)^{n} $$

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty} n^{2} / 2^{n} $$