Question

Find the Taylor Series centered at c=0 for f(x)=40x^(5)/(5+8x) and determine its interval of convergence

Answer 1

Do you know the formula for a Maclaurin series?

Answer 2

First you know that
\[
\frac1{1+ u}=\sum_{i=0}^\infty (-1)^n u^n, \quad |u| <1\\
\frac{ 40 x^5}{5+ 8 x}=\frac{ 40 x^5}{5(1+ \frac{8 x}5)}=\frac{ 8 x^5}{(1+ \frac{8 x}5)}\\

\frac{ 1}{1+ \frac{8 x}5}=\sum_{i=0}^\infty (-1)^n\left ( \frac {8x}5\right)^n, \quad |8x/5| <1\\
\]
You should be able to finish it now.

Answer 3

\[
\frac {1} {1 + \frac {8 x} 5} = \sum_ {i = 0}^\infty (-1)^n \frac {8^
n} {5^n} x^n, \quad | x | < 5/8 \\

\frac {8 x^5} {1 + \frac {8 x} 5} = \sum_ {i = 0}^\infty (-1)^n \frac {8^
{n+1}} {5^n} x^{n+5}, \quad | x | < 5/8 \\
\]