Question

Taylor Series Question ? Please help

Answer 1

Find the sum of \[\sum_{p=1}^{\infty} px ^{p-1} for |x| < 1\]

Answer 2

Please with steps

Answer 3

Notice that
\[\sum_{p=1}^\infty px^{p-1}=\frac{d}{dx}\sum_{p=0}^\infty x^p\]
The series on the RHS is a geometric series, which for \(|x|<1\) has the sum \(\dfrac{1}{1-x}\).

So,
\[\sum_{p=1}^\infty px^{p-1}=\frac{d}{dx}\left[\frac{1}{1-x}\right]=\cdots\]