91Ó°ÊÓ

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

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

Use the Limit Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{3}{2^{n}+3} $$

Find the sum of the given series. $$ \sum_{n=1}^{\infty} 7^{(-n / 3)} $$

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=3}^{\infty}(-1)^{n} \frac{1}{n \ln ^{2}(n)} $$

Find the sum of the given series. $$ \sum_{n=2}^{\infty} \frac{1}{2^{n / 2}} $$

Use the Limit Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{2^{n}+11}{3^{n}-1} $$

The given power series \(\sum_{n=0}^{\infty} a_{n}(x-c)^{n}\) defines a function \(f\). State the partial sum \(\sum_{n=0}^{4} b_{n}(x-c)^{n}\) of the power series for \(f^{\prime}(x),\) and determine the interval \(I\) of convergence of \(f^{\prime}(x)\). $$ \sum_{n=1}^{\infty} \frac{(x-1)^{n}}{n^{3 / 2}} $$

Compute the Taylor polynomial \(T_{N}\) of the given function \(f\) with the given base point \(c\) and given order \(N\). \(f(x)=\tan (x) \quad N=3 \quad c=0\)

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. $$ \sum_{n=8}^{\infty} \frac{e^{n}}{\left(1+e^{n}\right)^{2}} $$