Friday, 14 December 2018

sequences and series - Sum suminftyx=0fracx2x




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=11r



if r(1,1). This is the geometric series. If you haven't seen this proven before, here's a proof. Define



SN=Nn=0rn.



Then




rSN=Nn=0rn+1=N+1n=1rn=SN1+rN+1.



Solve this equation for SN, obtaining



SN=1rN+11r



and send N to conclude.



The sum above converges absolutely, so we can differentiate term by term. Doing so we get




f(r)=n=0nrn1=1(1r)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(1r)2.



Now substitute r=1/2.


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