Work: Typical Approach - f(x)=$xe^x$ f(1)=$e$ f'(x)=$e^x+xe^x$ f'(1)=$2e$ f''(x)=$2e^x+xe^x$ f''(1)=3e
Resulting Taylor Series: $e+2e(x-1)+3/2e(x-1)^2$
My Approach - Find the first degree Taylor Series for $e^x$ centered around a = 1
Let g(x)=$e^x$ $g^n(x)$ = $e^x$ Therefore all coefficients are e for a=1. Result: $e+e(x-1)$
To find the second degree polynomial of $xe^x$ simply multiply the previous Taylor series by x.
Result: $ex^2$ This comes about as the result of $\sum_{i=0}^\infty xe(x-1)^n/n!$
Justification: If a Taylor Series Approximation's Goal is to approximate a curve around a point, then my approach achieves the end result as the typical approach.
Typical: Green Mine: Blue image
If the image is not enough, WolframAlpha tests the series as a whole for equivalency to a function: equivalency
