91Ó°ÊÓ

State where the power series is centered. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} 1 \cdot 3 \cdot \cdots(2 n-1)}{2^{n} n !} x^{n} $$

Find a geometric power series for the function, centered at 0, (a) by the technique shown in Examples 1 and 2 and (b) by long division. $$ f(x)=\frac{1}{1+x} $$

Find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\tan x, \quad c=\frac{\pi}{4} $$

Find a power series for the function, centered at \(c\), and determine the interval of convergence. $$ f(x)=\frac{3}{2 x-1}, \quad c=2 $$

Use the Integral Test to determine the convergence or divergence of the \(p\) -series. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\)

Find the Maclaurin series for the function. \(f(x)=e^{x}+e^{-x}=2 \cosh x\)

Test for convergence or divergence, using each test at least once. Identify which test was used. (a) \(n\) th-Term Test (b) Geometric Series Test (c) \(p\) -Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test $$ \sum_{n=0}^{\infty} 5\left(-\frac{1}{5}\right)^{n} $$

Simplify the ratio of factorials. $$ \frac{(n+2) !}{n !} $$

Approximate the sum of the series by using the first six terms.\(\sum_{n=0}^{\infty} \frac{(-1)^{n} 2}{n !}\)

Use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit. $$ a_{n}=\cos \frac{n \pi}{2} $$