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Let $(a_n)_n \subset \mathbb R$ be a sequence of real numbers and consider the series $$ \sum_{n=1}^{\infty} a_nx^{1/n}. $$ Show that if the series converges for some $x_0>0$ then it converges in $(0,+\infty)$.

I was trying to use comparison test at least in the case $x<x_0$ but, as Jonathan's comment pointed out, the $a_n$ may not be positive.

So the problem is how to start: have you got any ideas? Any hints, please? Thank you.

user80551
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Romeo
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1 Answers1

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Hint: Given any $x>0$, note that $$\sum_{n\geq 1}a_nx^{1/n} = \sum_{n\geq 1}a_nx_0^{1/n}\sqrt[n]{\frac{x}{x_0}}$$

Now, let $b_n = a_nx_0^{1/n}$; $c_n = \sqrt[n]{\frac{x}{x_0}}$.

What tests do you know that would prove the convergence of $$\sum_{n\geq 1}b_nc_n?$$

Edit (additional hints) Let's recount what we know:

  1. The example of $a_n = \frac{(-1)^n}{n}$ (at, for example, $x_0=1$) shows that the series might converge conditionally (and not absolutely), so any test that would imply absolute convergence is out of the question.
  2. We know--as you've mentioned in comments--that $\sum_{n\geq 1}b_n$ converges.
  3. We know that $c_n\to 1$ (hence trivially implying $c_n\not\to 0$, which I only mention because of its importance in some tests). What's more, we know that this convergence is monotonic. We do not, however, know if $(c_n)$ is an increasing/decreasing sequence.

That's about everything of use I can think of, but it should suffice to point you to the convergence test I have in mind (not to imply other profitable approaches don't exist; I just can't think of any).

Edit (answer)

Abel's convergence test deals with just this sort of situation.

Jonathan Y.
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  • Maybe should I use this test: http://math.stackexchange.com/questions/85287/b-n-bounded-sum-a-n-converges-absolutely-then-sum-a-nb-n-also ? Thanks. – Romeo Sep 24 '13 at 15:59
  • @Romeo, you don't know that $\sum b_n$ converges absolutely. Think of convergence tests for series which might converge conditionally (as this is a very real possibility here). – Jonathan Y. Sep 24 '13 at 16:33
  • @Romeo, I've added some details. I hope that helps. – Jonathan Y. Sep 24 '13 at 16:49
  • I apologize for being late but I was busy today, sorry. Thank you for your hints, I understood what you meant without looking to your answer but simply reading your hints. Just for sake of completeness, I add that if $x<x_0$ then $c_n$ is increasing, while if $x>x_0$ the sequence $c_n$ is decreasing. In any case, it is monotonic (and trivially bounded, as it converges) hence we can apply Abel's theorem. Am I right? Thank you very much! – Romeo Sep 25 '13 at 22:02
  • @Romeo you've got it. – Jonathan Y. Sep 25 '13 at 23:42