91Ó°ÊÓ

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=3}^{\infty}\left(\frac{\ln (n)}{\ln \left(n^{2}-4\right)}\right)^{n}\)

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty}\left(\frac{37}{n}\right)^{n}\)

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. $$ \sum_{n=1}^{\infty} \frac{1}{n+3} $$

Find the sum of the given series. $$ \sum_{n=1}^{\infty} 8^{-n} $$

Compute the Taylor polynomial \(T_{N}\) of the given function \(f\) with the given base point \(c\) and given order \(N\). \(f(x)=\sin (3 x)^{2} \quad N=4 \quad c=\pi / 6\)

Find the open interval on which the given power series converges absolutely. $$ \sum_{n=1}^{\infty}\left(\frac{1}{n^{3}}-\frac{1}{3^{n}}\right)(x+1)^{n} $$

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=\ln \left(1+x^{2}\right) $$

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\arctan (n)}{2 n^{4}-n} $$

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

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=\sqrt{1+x^{3}} $$