Find a power series for f(x)=xe^x

Question

a.)Find a power series for f(x)=xe^x

 

b.) Using the Taylor Series you found in part a, approximate 2e^2 using the first five terms, rounding to two decimal places.

 

 

Answer 1

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

 

So, xe^x = x + x² + x³/2! + x⁴/3! + x⁵/4! + ...

 

Using the first 5 terms, 2e² ≈ 2 + 4 + 8/2 + 16/6 + 32/24

 

                                        = 14