# Sequence Limit: $\lim\limits_{n \rightarrow \infty}{n\,x^n}$

If $-1<x<1$ show that $\lim\limits_{n \to \infty}{n\,x^n} = 0$. I don't have idea. I only prove that $n\,x^n$ is decreasing.

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What methods do you know for finding the limit of a sequence? –  Eric Stucky Jul 20 '12 at 0:51
more basic possible. –  jon jones Jul 20 '12 at 1:12
Are you able to show that if $|r|<1$, then $r^n \to 0$? –  copper.hat Jul 20 '12 at 1:17
It's not true that $n x^n$ is decreasing, although it is true that $|n x^n|$ is decreasing for large enough $n$. –  Robert Israel Jul 20 '12 at 1:27
Your acceptation rate converges to zero faster than any other sequence I've ever met...perhaps you'd want to fix this? –  DonAntonio Jul 20 '12 at 2:48

## 5 Answers

Consider the series $\sum n x^n$. The ratio test shows that this series converges when $|x|<1$. Hence in this case the sequence of terms must converge to zero.

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But the series test is just the same for sequences. In fact, if $\lim\;\left| \dfrac{a_{n+1}}{a_n}\right|<1$ then $a_n \to 0$. –  Pedro Tamaroff Jul 20 '12 at 1:42
@Ihf, I like your answer but in order to be able to use it I think you must put $\,n|x|^n\,$, as the ratio series is for positive series. Anyway, +1 –  DonAntonio Jul 20 '12 at 2:50
@DonAntonio, but if a series converges absolutely, then it converges. See Wikipedia for this version of the ratio test. –  lhf Jul 20 '12 at 10:59
Oh, I know that, @Ihf, but the ratio test is appliable only to positive series, that's all. My point is that, since $\,-1<x<1\,$, the test cannot be applied to the series as it is. Of course, once you take $\,|nx^n|\,$ and prove convergence, you have absolute conv. and thus conv. –  DonAntonio Jul 20 '12 at 14:00

I believe that one could use the monotonicity of the functions of $n$ in order to justify taking the continuous derivative with respect to $n$ in the l'Hopital argument above (by oen). This would still only work for x nonnegative (by what James mentioned). But then, couldn't one argue that, because $|nx^n| = |ny^n|$ for positive integers $n$ and for constants $x$ and $y$ (where $x=-y$ and $y$ is positive), that $\lim_{n\to\infty}|nx^n| = \lim_{n\to\infty}|ny^n| = \lim_{n\to\infty}ny^n = 0$. (The second to last equality is valid because $ny^n >0$ and the last equality is the limit whose value is given by the l'Hopital argument.) Could one then say that since $\lim_{n\to\infty}|nx^n| = 0$ that $\lim_{n\to\infty}nx^n = 0$? (-1 < x < 0) This seems fine to me, but let me know what you think.

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If $x=0$ we're done. Otherwise, we can show absolute convergence using l'Hôpital's, $$\begin{eqnarray*} \lim_{n\to\infty}|n x^n| &=& \lim_{n\to\infty} \frac{n}{|x|^{-n}} \\ &=& \lim_{n\to\infty} \frac{\frac{d}{dn} n}{\frac{d}{dn} |x|^{-n}} \\ &=& \lim_{n\to\infty} \frac{1}{-|x|^{-n}\log |x|} \\ &=& \lim_{n\to\infty} -\frac{|x|^n}{\log |x|} \\ &=& 0. \end{eqnarray*}$$

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I don't think this is strictly correct for $x<0$. You differentiate with respect to $n$, treating it as a continuous variable. But for $x<0$, the quantity $x^n$ is not necessarily real for $n \not \in \mathbb{N}$. –  James Fennell Jul 20 '12 at 9:00
@JamesFennell: Thanks, James. I believe you are right. –  user26872 Jul 20 '12 at 14:55

Hint: if $|x| < r < 1$, show that $|(n+1) x^{n+1}| < r\, |n x^n|$ for sufficiently large $n$.

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If $-1<x<1$ then $x=\dfrac{1}{r}$, with $|r|>1$. For example:

$$1>0.1=\frac{1}{10}$$ $$1>0.25=\frac{1}{4}$$ $$1>0.\overline 3 =\frac{1}{3}$$

Then, we can write your sequence as

$$a_n=\frac{n}{r^n}$$

Can you try and see what would happen to $a_n$ for large $n$?

Say $r=2$. Then what would $$\lim\limits_{n\to \infty}\frac{n}{2^n}$$ be? Can you try and generalize?

Also note that for positive $r$ $$a_{n+1}=\frac{n+1}{r^{n+1}}=\frac{1}{r}\frac{n}{r^n}+\frac{1}{r}\frac{1}{r^n}=$$ $$=\frac{1}{r}a_n+\frac{1}{r}\frac 1 n a_n<\frac{1}{r}a_n+\frac{1}{r}a_n=\frac{2}{r}a_n$$

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Where you wrote, in your first line, "...for $\,r>1\,$" , it must be, I believe, "...for $\,r>1\,\,\,or\,\,\,r<-1\,$ –  DonAntonio Jul 20 '12 at 2:52
@DonAntonio That should have read $|r|>1$. Fixing... –  Pedro Tamaroff Jul 20 '12 at 2:54