Calculus II · Sequences, series, and convergence tests
A harmonic series whose dyadic signs still do not settle
Problem
For $n\ge1$, let $k(n)$ be the unique nonnegative integer such that \[ k(n)\le\log_2n<k(n)+1. \] Determine whether \[ \sum_{n=1}^{\infty}\frac{(-1)^{k(n)}}n \] converges. With $B_k=\sum_{n=2^k}^{2^{k+1}-1}1/n$, prove $B_k\to\ln2$ as a Riemann sum and use dyadic endpoint partial sums to prove divergence. Explain why pairing adjacent opposite-sign blocks cannot alone prove convergence of the original series.
Hint
The sign is constant and equal to (-1) to the k on dyadic block k.
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