Last month I was calculating ∫∞011+x4dx when I stumbled on the surprising identity:
∞∑n=0(−1)n(14n+1+14n+3)=π√8
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: n∑i=0ai=n∑i=−n−1(−1)i4i+1 This transformation allows me to then mimic the proof that was suggested in the comments by @user17762.
(n∑i=0ai)2−n∑i=0bi=(n∑i=−n−1(−1)i4i+1)2−n∑i=−n−11(4i+1)2=n∑i,j=−n−1i≠j(−1)i4i+1(−1)j4j+1=n∑i,j=−n−1i≠j(−1)i+j4j−4i(14i+1−14j+1)=n∑i,j=−n−1i≠j(−1)i+j2j−2i⋅14i+1=12n∑i=−n−1(−1)i4i+1n∑j=−n−1i≠j(−1)jj−i=12n∑i=−n−1(−1)i4i+1ci,n=12n∑i=0aici,n
Where the last equality follows from ci,n=c−i−1,n. Since ci,n is a partial alternating harmonic sum, it is bounded by its largest entry in the sum: |ci,n|≤1n−i+1. We also know that |ai|≤24i+1. Apply these two inequalities to get:
|(n∑i=0ai)2−n∑i=0bi|≤12n∑i=024i+1⋅1n−i+1≤n∑i=014n+5(44i+1+1n−i+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 m≥3, then:
(1+1m−1−1m+1−12m−1+12m+1+13m−1−⋯)2= (∞∑i=−∞(−1)iim+1)2= ∞∑i=−∞1(im+1)2= 1+1(m−1)2+1(m+1)2+1(2m−1)2+1(2m+1)2+1(3m−1)2+⋯= (πmsinπm)2
The last equality follows from the comment by @Lucian.
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