Q. 5 - Calculus

Question

If the series $\sum _{k=0}^{\infty }{c}_k{\left(x-x_0\right)}^k$ converges to the function $h\left(x\right)$ for every real number, provide a formula for $c_k$ in terms of the function $h$. 

Step-by-Step Solution

Answer

The formula for $c_k$ is $\frac{h^{\left(k\right)}\left(x_0\right)}{k!}$.

Step 1. Given Information.

The series $\sum _{k=0}^{\infty }{c}_k{\left(x-x_0\right)}^k$ converges to $h\left(x\right)$for every real number.

Step 2. Explanation.

The series is converges to the function so, Taylor polynomial for h(x) in the form 

$\sum _{k=0}^{\infty }\frac{h^{\left(k\right)}\left(x_0\right)}{k!}{\left(x-x_0\right)}^k$.

For series $\sum _{k=0}^{\infty }{c}_k{\left(x-x_0\right)}^k$,  $c_k=\frac{h^{\left(k\right)}\left(x_0\right)}{k!}$