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If a series converges does the third power of the series converge? I want to say that if the terms of the series are all >0 then it does but if the series terms can be positive or negative I am not sure.

CiaPan
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MathIsHard
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  • Third power applied on each term? – barak manos Sep 15 '16 at 15:02
  • Does the $\sum^{\infty}{j=1}(b_j)^3$ converge or diverge if $\sum^{\infty}{j=1}(b_j)$ convereges? Sorry for the confusion. – MathIsHard Sep 15 '16 at 15:04
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    See http://math.stackexchange.com/questions/96666/if-sum-1-inftya-n3-diverges-does-sum-1-inftya-n – sTertooy Sep 15 '16 at 15:06
  • There is an edit button below your question. Please edit in the clarification so people see it easily. – Ross Millikan Sep 15 '16 at 15:06
  • I think that $\sum\limits_{n=1}^{\infty}b_n$ converges $\implies \lim\limits_{n\to\infty}b_n=0\implies\sum\limits_{n=1}^{\infty}b_n^3$ converges. – barak manos Sep 15 '16 at 15:07
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    @barakmanos No, see the link above. This is true if the $b_n$'s are non-negative, but not in general. – Clement C. Sep 15 '16 at 15:07
  • Thanks for the help. Can you say that if all terms are positive and they converge that the third power converges? I see that if we have negative terms there are counter examples. – MathIsHard Sep 15 '16 at 15:16
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    @ryBear If all terms are non-negative (or positive a fortiori), then this becomes immediate by comparison. Then $b_n\xrightarrow[n\to\infty]{} 0$, so for $n$ big enough, $b_n^2 < 1$ and $0 \leq b_n^3 \leq 1\cdot b_n$. – Clement C. Sep 15 '16 at 15:18

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Hint:

$$1-2+1+1-2+1+1-2+1+\cdots$$ nearly converges, while

$$1-8+1+1-8+1+1-8+1+\cdots$$ not.

What if you divide every term by $\sqrt[3]n$ ?