Evaluate the taylor series sum: n=2(1)n3n1(2n)22n1 ?

n=2(1)n3n1(2n)22n1 use taylor series to evaluate this sum.

1 Answer
Jun 22, 2018

n=2(1)n(2n)3n122n1=3349

Explanation:

Start from the geometric series:

n=0qn=11q for |q|<1

Let q=x to have:

n=0(1)nxn=11+x for |x|<1

Differentiate term by term:

n=1(1)nnxn1=1(1+x)2 for |x|<1

Let x=34:

n=1(1)nn(34)n1=1(1+34)2

n=1(1)nn3n122n2=1649

n=1(1)n(2n)3n122n1=1649

Extract the term for n=1:

1+n=2(1)n(2n)3n122n1=1649

n=2(1)n(2n)3n122n1=1649+1

n=2(1)n(2n)3n122n1=3349