Harmony Oneal

2022-11-25

Suppose $\sum _{n=1}^{\mathrm{\infty }}{x}_{n}<\mathrm{\infty }$. Then prove that $\sum _{n=1}^{\mathrm{\infty }}{x}_{n}{y}_{n}$ converges.

Cale Terry

Expert

Let $N\ge 2$ then by telescoping
${y}_{N}={y}_{1}-\sum _{n=1}^{N-1}\left({y}_{n}-{y}_{n+1}\right)$
hence the sequence $\left({y}_{N}\right)$ is convergent since
$\underset{N\to \mathrm{\infty }}{lim}\sum _{n=1}^{N-1}\left({y}_{n}-{y}_{n+1}\right)=\sum _{n=1}^{\mathrm{\infty }}\left({y}_{n}-{y}_{n+1}\right)<\mathrm{\infty }$
by M>0 and by
$inf\left(M{x}_{n},-M{x}_{n}\right)\le {x}_{n}{y}_{n}\le max\left(M{x}_{n},-M{x}_{n}\right)$
and $\sum _{n=1}^{\mathrm{\infty }}{x}_{n}$ is convergent we have the result.

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