- #1
MAGNIBORO
- 106
- 26
hi, I'm solving solving a problem about sums of zeta function and I'm come to the following conclusion
$$\sum _{n=2}^{\infty }{\frac {\zeta \left( n \right) }{{k}^{n}}}=
\sum _{s=1}^{\infty } \left( {\it ks} \left( {\it ks}-1 \right)
\right) ^{-1}=\int_{0}^{1}\!{\frac {{u}^{k-2}}{\sum _{i=0}^{k-1}{u}^{i}}}\,{\rm d}u,\: \:\: \: \forall k, \: \:k\geq 2 \: \: and \: \: k\in \mathbb{N}$$
and from wolfram alpha tell me that
$$=-{\frac {1}{k} \left( \psi \left( 1-{k}^{-1} \right) +\gamma \right) }$$
where the ##\psi## is the "zero" digamma function (I don't know how It is said )
I don't know what is that true and I would like someone to show me this relation. thx
$$\sum _{n=2}^{\infty }{\frac {\zeta \left( n \right) }{{k}^{n}}}=
\sum _{s=1}^{\infty } \left( {\it ks} \left( {\it ks}-1 \right)
\right) ^{-1}=\int_{0}^{1}\!{\frac {{u}^{k-2}}{\sum _{i=0}^{k-1}{u}^{i}}}\,{\rm d}u,\: \:\: \: \forall k, \: \:k\geq 2 \: \: and \: \: k\in \mathbb{N}$$
and from wolfram alpha tell me that
$$=-{\frac {1}{k} \left( \psi \left( 1-{k}^{-1} \right) +\gamma \right) }$$
where the ##\psi## is the "zero" digamma function (I don't know how It is said )
I don't know what is that true and I would like someone to show me this relation. thx