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Prove the limit and find the value on zeta function

Source: 2019 Jozsef Wildt International Math Competition-W. 30

May 18, 2020
zeta functionlimitSummationfunction

Problem Statement

[*] Prove that lim⁡n→∞(n+14−ζ(3)−ζ(5)−⋯−ζ(2n+1))=0\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0 [*] Calculate ∑n=1∞(n+14−ζ(3)−ζ(5)−⋯−ζ(2n+1))\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)
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