Question

find the taylor series of 5/(1-5x)^2

Answer 1

Consider the antiderivative,
\[\int\frac{5}{(1-5x)^2}\,dx=\frac{1}{1-5x}+C\]
Recall that for \(|x|<1\), you have
\[\frac{1}{1-x}=\sum_{k=0}^\infty x^k\]
So, if \(f(x)=\dfrac{5}{(1-5x)^2}\), and \(F(x)=\displaystyle\int f(x)\,dx\), then
\[F(x)=\sum_{k=0}^\infty (5x)^k+C\]
Differentiating will give you the series for \(f(x)\).

Answer 2

thanks, do u have to change the starting index k to 1 also?

Answer 3

ok i think so :)

Answer 4

It wouldn't matter.
\[\sum_{k=0}^\infty k x^{k-1}=0+\underbrace{1+2x+3x^2+\cdots}_{\text{starting at }k=1}=\sum_{k=1}^\infty kx^{k-1}\]

Answer 5

kk i see :D