6.1 Power series and functions  (Page 5/8)

${\displaystyle \sum _{n=1}^{\infty }\frac{n{x}^n}{e^n}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{n^2{x}^n}{2^n}}$

${\displaystyle \sum _{k=1}^{\infty }\frac{k^e{x}^k}{e^k}}$

${\displaystyle \sum _{k=1}^{\infty }\frac{{\pi }^k{x}^k}{k^{\pi }}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{n\text{!}}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{{10}^n{x}^n}{n\text{!}}}$

${\displaystyle \sum _{n=1}^{\infty }{\left(\mathrm{-1}\right)}^n}\frac{x^n}{\text{ln}\left(2n\right)}$

${\displaystyle \sum _{k=1}^{\infty }\frac{{\left(k\text{!}\right)}^2{x}^k}{\left(2k\right)\text{!}}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{\left(2n\right)\text{!}{x}^n}{n^{2n}}}$

${\displaystyle \sum _{k=1}^{\infty }\frac{k\text{!}}{1·3·5\text{⋯}\left(2k-1\right)}}{x}^k$

${\displaystyle \sum _{k=1}^{\infty }\frac{2·4·6\text{⋯}2k}{\left(2k\right)\text{!}}{x}^k}$

${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{\left(\begin{array}{c}2n\\ n\end{array}\right)}}$ where $\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n\text{!}}{k\text{!}\left(n-k\right)\text{!}}$

${\displaystyle \sum _{n=1}^{\infty }{\text{sin}}^{2}n{x}^n}$

${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(n\text{!}\right)}^3}{\left(3n\right)\text{!}}{x}^n}$

${\displaystyle \sum _{n=1}^{\infty }\frac{2^{3n}{\left(n\text{!}\right)}^3}{\left(3n\right)\text{!}}{x}^n}$

${\displaystyle \sum _{n=1}^{\infty }\frac{n\text{!}}{n^n}{x}^n}$

${\displaystyle \sum _{n=1}^{\infty }\frac{\left(2n\right)\text{!}}{n^{2n}}{x}^n}$

$f\left(x\right)=\frac{1}{x};a=1$ ( Hint: $\frac{1}{x}=\frac{1}{1-\left(1-x\right)})$

$f\left(x\right)=\frac{1}{1-x^2};a=0$

$f\left(x\right)=\frac{x}{1-x^2};a=0$

$f\left(x\right)=\frac{1}{1+x^2};a=0$

$f\left(x\right)=\frac{x^2}{1+x^2};a=0$

$f\left(x\right)=\frac{1}{2-x};a=1$

$f\left(x\right)=\frac{1}{1-2x};a=0.$

$f\left(x\right)=\frac{1}{1-4x^2};a=0$

$f\left(x\right)=\frac{x^2}{1-4x^2};a=0$

$f\left(x\right)=\frac{x^2}{5-4x+x^2};a=2$

Explain why, if ${\left|a_n\right|}^{1\text{/}n}\to r>0,$ then ${\left|a_n{x}^n\right|}^{1\text{/}n}\to \left|x\right|r<1$ whenever $\left|x\right|<\frac{1}{r}$ and, therefore, the radius of convergence of ${\displaystyle \sum _{n=1}^{\infty }{a}_n{x}^n}$ is $R=\frac{1}{r}.$

${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{n^n}}$

${\displaystyle \sum _{k=1}^{\infty }{\left(\frac{k-1}{2k+3}\right)}^k{x}^k}$

${\displaystyle \sum _{k=1}^{\infty }{\left(\frac{2k^2-1}{k^2+3}\right)}^k{x}^k}$

${\displaystyle \sum _{n=1}^{\infty }{a}_n}={\left(n^{1\text{/}n}-1\right)}^n{x}^n$

Suppose that $p\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ such that $a_n=0$ if n is even. Explain why $p\left(x\right)=p\left(\text{−}x\right).$

Suppose that $p\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ such that $a_n=0$ if n is odd. Explain why $p\left(x\right)=\text{−}p\left(\text{−}x\right).$

Suppose that $p\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ converges on $\left(\mathrm{-1},1\right].$ Find the interval of convergence of $p\left(Ax\right).$

Suppose that $p\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ converges on $\left(\mathrm{-1},1\right].$ Find the interval of convergence of $p\left(2x-1\right).$

${\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^{2n}}$

${\displaystyle \sum _{n=0}^{\infty }{a}_{2n}{x}^{2n}}$

${\displaystyle \sum _{n=0}^{\infty }{a}_{2n}{x}^n}\left(Hint\text{:}x=\text{±}\sqrt{x^2}\right)$

${\displaystyle \sum _{n=0}^{\infty }{a}_{n^2}{x}^{n^2}}$ ( Hint: Let $b_k=a_k$ if $k=n^2$ for some n , otherwise $b_k=0.)$

Suppose that $p\left(x\right)$ is a polynomial of degree N . Find the radius and interval of convergence of ${\displaystyle \sum _{n=1}^{\infty }p\left(n\right)x^n}.$

[T] Plot the graphs of $\frac{1}{1-x}$ and of the partial sums $S_N={\displaystyle \sum _{n=0}^N{x}^n}$ for $n=10,20,30$ on the interval $\left[\mathrm{-0.99},0.99\right].$ Comment on the approximation of $\frac{1}{1-x}$ by $S_N$ near $x=\mathrm{-1}$ and near $x=1$ as N increases.

[T] Plot the graphs of $\text{−}\text{ln}\left(1-x\right)$ and of the partial sums $S_N={\displaystyle \sum _{n=1}^N\frac{x^n}{n}}$ for $n=10,50,100$ on the interval $\left[\mathrm{-0.99},0.99\right].$ Comment on the behavior of the sums near $x=\mathrm{-1}$ and near $x=1$ as N increases.

[T] Plot the graphs of the partial sums $S_n={\displaystyle \sum _{n=1}^N\frac{x^n}{n^2}}$ for $n=10,50,100$ on the interval $\left[\mathrm{-0.99},0.99\right].$ Comment on the behavior of the sums near $x=\mathrm{-1}$ and near $x=1$ as N increases.

[T] Plot the graphs of the partial sums $S_N={\displaystyle \sum _{n=1}^N\text{sin}n{x}^n}$ for $n=10,50,100$ on the interval $\left[\mathrm{-0.99},0.99\right].$ Comment on the behavior of the sums near $x=\mathrm{-1}$ and near $x=1$ as N increases.

[T] Plot the graphs of the partial sums $S_N={\displaystyle \sum _{n=0}^N{\left(\mathrm{-1}\right)}^n\frac{x^{2n+1}}{\left(2n+1\right)\text{!}}}$ for $n=3,5,10$ on the interval $\left[\mathrm{-2}\pi ,2\pi \right].$ Comment on how these plots approximate $\text{sin}x$ as N increases.

[T] Plot the graphs of the partial sums $S_N={\displaystyle \sum _{n=0}^N{\left(\mathrm{-1}\right)}^n\frac{x^{2n}}{\left(2n\right)\text{!}}}$ for $n=3,5,10$ on the interval $\left[\mathrm{-2}\pi ,2\pi \right].$ Comment on how these plots approximate $\text{cos}x$ as N increases.