Q. 4 - Calculus

Question

What is a power series in $x-x_0$?

Step-by-Step Solution

Answer

A power series in $x-x_0$ is $\sum _{k=0}^{\infty }{a}_k{\left(x-x_0\right)}^k=a_0+a_1\left(x-x_0\right)+a_2{\left(x-x_0\right)}^2+...$

Step 1. Given information.

Consider the given question,

$x-x_0$.

Step 2. Consider a power series.

If $\sum _{k=0}^{\infty }{a}_k$ is a sequence of real numbers and x is a variable and $x_0$ be a real number. Then a power series in $x-x_0$ is a series of the form $\sum _{k=0}^{\infty }{a}_k{\left(x-x_0\right)}^k$.

Rewriting the series,

$\sum _{k=0}^{\infty }{a}_k{\left(x-x_0\right)}^k=a_0+a_1\left(x-x_0\right)+a_2{\left(x-x_0\right)}^2+...$