Q. 6 - Calculus

Question

 Let $f(x$) be a function such that the power series in $x-x_0,$ $\sum _{k-0}^{\infty }{a}_k{\left(x-x_0\right)}^k$ converges absolutely to $f$ on the interval I. If $G_1$ and $G_2$ are two antiderivatives for $f$, explain why the power series in $x-x_0$ for $G_1$and $G_2$ have the same interval of convergence.

Step-by-Step Solution

Answer

Hence, if $G_1$ and $G_2$are the two antiderivatives for f, then the power series in $x-x_0$ for $G_1$ and $G_2$ have the same interval of convergence.

step 1: given information

 Consider the power series $f\left(x\right)=\sum _{k=0}^{\infty }{a}_k{\left(x-x_0\right)}^k$ converges absolutely to $f$ on the interval $I$$l .$

step 2: calculation

Now, if $G_1$ and $G_2$  are the two antiderivatives for f , then the must differ only in the constants Also, while calculating the interval of convergence, we do not have any effect of constants on the interval of convergence of the power series.

Hence, if $G_1$ and $G_2$ are the two antiderivatives for $f$, then the power series in $x-x_0$ for $G_1$ and $G_2$ have the same interval of convergence.