I'm studying for an exam and I ran into these problems. I'm having a feeling that this is not true. Hence, I don't need to prove. I need to just provide a counterexample. However, the appropriate example is not just coming. Any help?
Prove or give a counterexample:
If ∑∞n=1an=1 and each an≥0, then lim
If a_n\geq 0 and \sum_{n=1}^{\infty}a_n converges, then \lim\limits_{n\to\infty}na_n=0
Answer
Let a_{n^2}=\dfrac{1}{n^2} and a_n=0 otherwise. Then na_n does not tend to 0, and the series is convergent.
The sum is \dfrac{\pi^2}{6}, but note you can multiply each term by the same constant to make the limit any nonzero number you wish.
Note, however, that if a_n is also nonincreasing, the situation is different: Series converges implies \lim{n a_n} = 0
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