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n-2-n-n-2-n-




Question Number 158522 by amin96 last updated on 05/Nov/21
Σ_(n=2) ^∞ (((ζ(n)−n))/2^n )=?
n=2(ζ(n)n)2n=?
Answered by qaz last updated on 06/Nov/21
Σ_(n=2) ^∞ ((ζ(n)−n)/2^n )  =Σ_(n=2) ^∞ Σ_(k=1) ^∞ (1/((2k)^n ))−(3/2)  =Σ_(k=1) ^∞ ((1/((2k)^2 ))/(1−(1/(2k))))−(3/2)  =(1/2)Σ_(k=1) ^∞ ((1/(k−(1/2)))−(1/k))−(3/2)  =−(1/2)H_(−1/2) −(3/2)  =ln2−(3/2)
n=2ζ(n)n2n=n=2k=11(2k)n32=k=11(2k)2112k32=12k=1(1k121k)32=12H1/232=ln232
Commented by amin96 last updated on 06/Nov/21
thanks sir. very good solution
thankssir.verygoodsolution

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