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Calculus and Beyond Homework Help
If a series converges with decreasing terms, then na_n -> 0
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[QUOTE="Mr Davis 97, post: 6068228, member: 515461"] [h2]Homework Statement [/h2] Prove that if ##(a_n)## is a decreasing sequence of positive numbers and ##\sum a_n## converges, then ##\lim na_n = 0## [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] Let ##\epsilon >0##. By the Cauchy criterion there exists an ##N\in \mathbb{N}## such that ##\forall n\ge m\ge N##, we have that ##|\sum_{k=m+1}^{n}a_k|<\epsilon##. But the sequence is decreasing, so ##|(n-m)a_n|\le |\sum_{k=m+1}^{n}a_k|<\epsilon##. So we have that ##|na_n-ma_n|<\epsilon## for all ##\epsilon>0##. So ##na_n=ma_n## for all ##n\ge m\ge N##. Since the tails of these sequence are the same eventually, they have the same limit. Since ##\sum a_n## converges, we see that ##\lim ma_n = 0##. Hence ##\lim n a_n = 0##. [/QUOTE]
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Calculus and Beyond Homework Help
If a series converges with decreasing terms, then na_n -> 0
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