91Ó°ÊÓ

The given power series \(\sum_{n=0}^{\infty} a_{n}(x-c)^{n}\) defines a function \(f\). Calculate the partial sum \(\sum_{n=1}^{4} b_{n}(x-c)^{n}\) of the power series for \(F(x)=\int_{c}^{x} f(t) d t,\) and determine the interval \(I\) of convergence of \(F(x)\). $$ \sum_{n=0}^{\infty} \sqrt{n+1} x^{n} $$

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n+3^{n}}{n^{3}+2^{n}}\)

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. $$ \sum_{n=2}^{\infty} \frac{3}{n^{2}+n} $$

In each of Exercises \(39-42,\) compute the Taylor polynomial \(T_{3}(x)\) of the given function \(f\) with the given base point \(c\). \(f(x)-x \cdot e^{x} \quad c-0\)

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

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. $$ \sum_{n=4}^{\infty} \frac{1}{n \ln ^{4}(n)} $$

Find the sum of the given series. $$ \sum_{n=3}^{\infty}\left(2^{-n}+3^{-n}\right) $$

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

A convergent alternating series \(S=\sum_{n=1}^{\infty}(-1)^{n+1} a_{n}\) is given. Use inequality (8.4.1) to find a value of \(N\) such that the partial sum \(S_{N}=\sum_{n=1}^{N}(-1)^{n+1} a_{n}\) approximates the given infinite series to within 0.01 . That is, find \(N\) so that \(\left|S-S_{N}\right| \leq 0.01\) holds. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n !} $$

In each of Exercises \(41-44,\) use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{4^{n}}{\left(n^{1 / n}+2\right)^{n}}\)