91Ó°ÊÓ

In Exercises \(1-16,\) determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\)

In Exercises \(1-8,\) write the first five terms of the sequence. \(a_{n}=2^{n}\)

In Exercises \(1-6,\) find the first five terms of the sequence of partial sums. $$ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdot \cdot $$

In Exercises 1 and 2 , state where the power series is centered. $$ \sum_{n=0}^{\infty} n x^{n} $$

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. $$ f(x)=\frac{1}{2-x} $$

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? $$ f(x)=\frac{4}{\sqrt{x}}, \quad c=1 $$

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n+1} $$

In Exercises \(1-10,\) use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=e^{2 x}, \quad c=0 $$

In Exercises 1 and 2 , state where the power series is centered. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} $$

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. $$ f(x)=\frac{1}{1+x} $$