The series is 1⋅12+2⋅14+3⋅18+⋯
Or in other words
∞∑n=1n2n
What kind of series is this and how to find the sum? Thanks....
Answer
Without calculus:
If
s(a)=∑∞n=0nan
for
|a|<1,
then
as(a)=∑∞n=0nan+1=∑∞n=1(n−1)an=∑∞n=1nan−∑∞n=1an=s(a)−a1−asos(a)=a(1−a)2
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