Question

Find the taylor series for xex about x = 0. then integrate term-by-term and substitute to show that the series

Answer 1

The Taylor series for e^x about x = 0 is

1 + x + x^2/2! + x^3/3! + x^4 /4!+...

xe^x  = x + x^2 + x^3/2! + x^4/3! + x^5/4!+....

Integrating term by term gives:-

x^2/ 2 + x^3/3 + x^4 / 3.2!  + x^5 / 4.3! + x^6 /  5.4!+....