MathDB

Let

(t_n)_{n\geqslant 1}

be the sequence defined by

t_1=1, t_{2k}=-t_k

and

t_{2k+1}=t_{k+1}

for all

k\geqslant 1.

Consider the series

\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.

Prove that this series converges to a positive real number.
Proposed by Dylan Toh

Problem Statement