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Question Number 140506 by Dwaipayan Shikari last updated on 08/May/21

e^{(\frac{ζ (2)}{2} - \frac{ζ (3)}{3} + \frac{ζ (4)}{4} - \frac{ζ (5)}{5} + \dots)} = ?

Answered by mnjuly1970 last updated on 08/May/21

Commented by mnjuly1970 last updated on 08/May/21

H i m r p a y a n . .

p l e a s e c h e c k m y s o l u t i o n \dots .

Commented by Dwaipayan Shikari last updated on 08/May/21

C o r r e c t s i r! G r e a t!

Commented by Niiicooooo last updated on 08/May/21

n i c e p r o o f!

Commented by mnjuly1970 last updated on 08/May/21

m e r c e y \dots

Answered by Dwaipayan Shikari last updated on 08/May/21

ψ (z + 1) = ψ (1) + \frac{ψ^{'} (1)}{1!} z + \frac{ψ^{″} (1)}{2!} z^{2} + \dots

\Rightarrow ψ (z + 1) = - γ + ζ (2) z - ζ (3) z^{2} + ζ (4) z^{3} - \dots

\Rightarrow \int_{0}^{1} ψ (z + 1) d z = - γ + \int_{0}^{1} ζ (2) z - ζ (3) z^{2} + . . d z

\Rightarrow {[l o g (Γ (z + 1))]}_{0}^{1} + γ = \frac{ζ (2)}{2} - \frac{ζ (3)}{3} + \frac{ζ (4)}{4} - . .

γ = \frac{ζ (2)}{2} - \frac{ζ (3)}{3} + \dots

Commented by mnjuly1970 last updated on 08/May/21

n i c e v e r y n i c e \dots