Theorem
25
[21F] Let \(\{ a_{n}\} \) and \(\{ b_{n}\} \) be two sequences. If \( b_{n}\) tends monotonically to \(0\) and if the series of partial sums of \(a_ n\) is bounded, i.e. if
\[ b_{n}β₯ b_{n+1} {\gt}0\quad ,\quad \lim _{nββ } b_{n} = 0 \quad ,\quad β M{\gt}0,~ β Nββ~ , \left|β _{n=1}^{N}a_{n}\right|{\lt}M\quad , \]
then the series
\[ β _{n=1}^{+β }{a_{n}b_{n}} \]
is convergent.
The proof is left as an exercise (Hint: use [21H])
Solution
1