91Ó°ÊÓ

Find the Taylor polynomial \(T_{n}(x)\) for the function \(f\) at the number a. Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=x e^{-2 x}, \quad a=0$$

Test the series for convergence or divergence. $$\sum_{k=1}^{\infty} k^{2} e^{-k}$$

\(3-10\) Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{1+x}{1-x} $$

(a) Explain the difference between $$\sum_{i=1}^{n} a_{i} \quad \text { and } \quad \sum_{j=1}^{n} a_{j}$$ (b) Explain the difference between $$\sum_{i=1}^{n} a_{i} \quad \text { and } \quad \sum_{i=1}^{n} a_{j}$$

\(2-20\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{n}}{1+2 \sqrt{n}} $$

\(2-28\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\sqrt{n^{3}+2}}$$

Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty}\left(n^{-1.4}+3 n^{-1.2}\right)$$

Find the Taylor polynomial \(T_{n}(x)\) for the function \(f\) at the number a. Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=\tan ^{-1} x, \quad a=1$$

\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{n^{2}-1}{3 n^{4}+1}$$

Find the Maclaurin series for \(f(x)\) using the definition of a Maclaurin series. [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1\) Also find the associated radius of convergence. \(f(x)=x e^{x}\)