Real Analysis I · Sequences & Series
Comparison using bounded partial sums
Problem
Suppose the partial sums $A_n=\sum_{k=1}^na_k$ are bounded, while $b_n$ decreases to $0$. Prove $\sum a_nb_n$ converges.
Hint
Summation by parts gives $\sum_{k=p}^qa_kb_k=A_qb_q-A_{p-1}b_p+\sum_{k=p}^{q-1}A_k(b_k-b_{k+1})$.
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