Question

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{x}}^{\mathbf{n}}}{{\mathbf{(}\mathbf{n}\mathbf{!}\mathbf{)}}^{\mathbf{2}}}$

Short answer

All power series convergence intervals are X values.

Step-by-step solution

Significance of ratio test

The ratio test as follows:

Solution steps:

First,find ${\mathbf{\rho }}_{\mathbf{n}}\mathbf{=}\mathbf{\left|}\frac{{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}}{{\mathbf{a}}_{\mathbf{n}}}\mathbf{\right|}\mathbf{.}$

Second,$\mathbf{\rho }\mathbf{=}\underset{\mathbf{n}\mathbf{\to }\mathbf{\infty }}{\mathbf{lim}}\mathbf{}{\mathbf{p}}_{\mathbf{n}}\mathbf{.}$ .

Third,$\mathbf{p}\mathbf{<}\mathbf{1}$ t he series converges, if$\mathbf{p}\mathbf{>}\mathbf{1}$ the series diverges.

Finding convergence

Let us consider the power series

$\sum _{\mathrm{n}=1}^{\infty }\frac{{\mathrm{x}}^{\mathrm{n}}}{{(\mathrm{n}!)}^2}$

And we would like to find the interval of convergence of the power series.

We will be using the ratio test, as follows:

Solution steps:

First,find role="math" localid="1658820727600" ${\mathrm{\rho }}_{\mathrm{n}}=\left|\frac{{\mathrm{a}}_{\mathrm{n}+1}}{{\mathrm{a}}_{\mathrm{n}}}\right|$.

Second, $\mathrm{\rho }=\underset{\mathrm{n}\to \infty }{\lim }{\mathrm{p}}_{\mathrm{n}}.$

Third, $\mathrm{p}<1$the series converges, if $\mathrm{p}>1$the series diverges.

role="math" localid="1658820961551" $$\begin{gathered} {\mathrm{\rho }}_{\mathrm{n}}=\left|\frac{{\mathrm{a}}_{\mathrm{n}+1}}{{\mathrm{a}}_{\mathrm{n}}}\right| \\ =\left|\frac{{\displaystyle \frac{x^{n+1}}{\left(\right(n+1)!{)}^2}}}{{\displaystyle \frac{x^n}{(n!{)}^2}}}\right| \\ {\rho }_n=\left|\frac{\left(x^{n+1}\right(n!{)}^2}{\left(\right(n+1)!{)}^2{x}^n}\right| \end{gathered}$$

Finding the points of divergence

Notice that$(\mathrm{n}+1)!=(\mathrm{n}+1)\mathrm{n}!$ . Hence

$$\begin{gathered} {\mathrm{\rho }}_{\mathrm{n}}=\left|\frac{\mathrm{x}}{(\mathrm{n}+1{)}^2}\right| \\ \mathrm{\rho }=\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{\mathrm{x}}{(\mathrm{n}+1{)}^2}\right| \\ =0 \end{gathered}$$

$p<1$For all the values of x .

Concluding statement

All power series convergence intervals are x values.