Question

Perform the following steps for the power series in $x-x_0$in the given exercises:

(a) Find the interval of convergence, I, for the series.

(b) Let f be the function to which the series converges on I. Find the power series in $x-x_0$for f.

(c) Find the power series in $x-x_0\text{for}F\left(x\right)={\int }_{x_0}^x f\left(t\right)dt$.

$\sum _{k=0}^{\mathrm{\infty }} (-1{)}^k\frac{k!}{\left(2k\right)!}(x-7{)}^{2k}$

Short answer

Ans:

Step-by-step solution

Step 1. Given information.

given,

$\sum _{k=0}^{\mathrm{\infty }} (-1{)}^k\frac{k!}{\left(2k\right)!}(x-7{)}^{2k}$

Step 2.  (a)  Consider the power series ∑k=0∞ (−1)kk!(2k)!(x−7)2k

To find the eee