91Ó°ÊÓ

Write the first five terms of the sequence. \(a_{n}=\left(-\frac{1}{2}\right)^{n}\)

Find the first five terms of the sequence of partial sums. $$ 3-\frac{9}{2}+\frac{27}{4}-\frac{81}{8}+\frac{243}{16}-\cdots $$

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{1}{2-x}, \quad c=5 $$

In Exercises \(1-4,\) 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? $$ f(x)=\sec x, \quad c=\frac{\pi}{4} $$

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} n e^{-n / 2} $$

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\ln (n+1)} $$

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=\sin x, \quad c=\frac{\pi}{4} $$

Find the first five terms of the sequence of partial sums. $$ \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\cdot \cdot $$

In Exercises \(1-4,\) 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. $$ f(x)=\frac{4}{5-x}, \quad c=-2 $$