Saturday, February 13, 2016

real analysis - Show (1+frac13frac15frac17+frac19+frac111cdots)2=1+frac19+frac125+frac149+cdots



Last month I was calculating 011+x4dx when I stumbled on the surprising identity:




n=0(1)n(14n+1+14n+3)=π8



and I knew



n=01(2n+1)2=π28



So if I could find a proof that (n=0(1)n(14n+1+14n+3))2=n=01(2n+1)2 then this could be a new proof that ζ(2)=π26. I've thought over this for almost a month and I'm no closer on showing this identity.



Note: Article on the multiplication of conditionally convergent series: http://www.jstor.org/stable/2369519


Answer




Let ak=(1)k(14k+1+14k+3) and bk=1(4k+1)2+1(4k+3)2. The goal is to show that: (i=0ai)2=i=0bi
The key observation that I missed on my previous attempt is that: ni=0ai=ni=n1(1)i4i+1 This transformation allows me to then mimic the proof that was suggested in the comments by @user17762.
(ni=0ai)2ni=0bi=(ni=n1(1)i4i+1)2ni=n11(4i+1)2=ni,j=n1ij(1)i4i+1(1)j4j+1=ni,j=n1ij(1)i+j4j4i(14i+114j+1)=ni,j=n1ij(1)i+j2j2i14i+1=12ni=n1(1)i4i+1nj=n1ij(1)jji=12ni=n1(1)i4i+1ci,n=12ni=0aici,n
Where the last equality follows from ci,n=ci1,n. Since ci,n is a partial alternating harmonic sum, it is bounded by its largest entry in the sum: |ci,n|1ni+1. We also know that |ai|24i+1. Apply these two inequalities to get:
|(ni=0ai)2ni=0bi|12ni=024i+11ni+1ni=014n+5(44i+1+1ni+1)14n+5(5+ln(4n+1)+ln(n+1))0  as  n
This concludes the proof. In fact, with the same idea, you can prove this general family of identities: Fix an integer m3, then:



(1+1m11m+112m1+12m+1+13m1)2= (i=(1)iim+1)2= i=1(im+1)2= 1+1(m1)2+1(m+1)2+1(2m1)2+1(2m+1)2+1(3m1)2+= (πmsinπm)2
The last equality follows from the comment by @Lucian.


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