If ∑an is convergent and bn is a monotone bounded sequence
then ∑anbn is convergent.
What I did is
If bn is a monotone bounded sequence then bn is convergent and suppose that bn→B when b→∞.
The algebraic limit theorems says that ∑anB is convergent. Since 0≤anbn≤anB then by the comparison test ∑anbn is also convergent.
It's a wrong use of comparison test?
Answer
It is enough to apply Dirichlet's test. If ∑n≥1an is convergent to A then its partial sums are bounded, and if {bn}n≥1 is monotonic and bounded it is convergent to some B and B−bn is decreasing to zero. It follows that
∑n≥1anbn=AB−∑n≥1an(B−bn)⏟convergent by Dirichlet
is convergent as well.
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