Calculate ∞∑x=0x2x
So, this series converges by ratio test. How do I find the sum? Any hints?
Answer
As a first step, let us prove that
f(r):=∞∑n=0rn=11−r
if r∈(−1,1). This is the geometric series. If you haven't seen this proven before, here's a proof. Define
SN=N∑n=0rn.
Then
rSN=N∑n=0rn+1=N+1∑n=1rn=SN−1+rN+1.
Solve this equation for SN, obtaining
SN=1−rN+11−r
and send N→∞ to conclude.
The sum above converges absolutely, so we can differentiate term by term. Doing so we get
f′(r)=∞∑n=0nrn−1=1(1−r)2.
(Precisely speaking, the sum in the middle is ill-defined at r=0, in that it has the form 0/0. However, f′(0)=1 still holds. This doesn't matter for this problem, but it should be noted regardless.) Now multiply by r to change it into your form:
∞∑n=0nrn=r(1−r)2.
Now substitute r=1/2.
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