Q. 16 - Calculus

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 $x-x_0\text{ for }F\left(x\right)={\int }_{x_0}^x f\left(t\right)dt$.

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

Step-by-Step Solution

Answer

Ans: 

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 ) k k ! ( 2 k ) ! ( x − 7 ) 2 k

To find the eee