Prove that if ∑ b_k converges; {b} not necessarily monotonic, ∑ b is bounded?
Update:
Abel's convergence test relies on Direchlet's test for its proof. Direchlet's test requires all partial sums of a particular series to be bounded. Abel's test only specifies that it converges
More Questions From This User See All
Answers & Comments
Let assume (wrong assumsion) that if ∑ b_k converges, then {b} necessarily monotonic.
Lets look on the series: b_k = [(-1)^k]/k, this series is not monotonic and it converges.
=> the assumsion is wrong => if ∑ b_k converges, then {b} not necessarily monotonic.
b must be bounded.