Sunday, 4 December 2016

real analysis - Show that sumanbn is convergent




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 bnB when b.



The algebraic limit theorems says that anB is convergent. Since 0anbnanB 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 n1an is convergent to A then its partial sums are bounded, and if {bn}n1 is monotonic and bounded it is convergent to some B and Bbn is decreasing to zero. It follows that



n1anbn=ABn1an(Bbn)convergent by Dirichlet



is convergent as well.


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