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Does the following sequnce converge or diverge? $$\sum_{n=1}^\infty\frac{n}{3^n}$$

How are these kinds of problems written down and proven? From my understanding, this will converge to $0$ because the denominator grows at a faster rate than the numerator. Is that all I can do about this problem or is there some more detailed way to solve it?

user737163
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    Certainly the terms of the series will converge to $0$ (essentially for the reason you stated), but the series itself will not tend to $0$, as it consists only of strictly positive terms. Unfortunately, knowing the series terms converge to $0$ is not enough to conclude convergence of the sum. Are you familiar with the ratio test? – Theo Bendit Mar 21 '21 at 16:06
  • @TheoBendit No. Only one week of classes passed and we just started sequences. Possibly this is all that is expected from me then, maybe. I wanted to check if there was some easy proof that I missed. – user737163 Mar 21 '21 at 16:08
  • You are asking about the convergence of a series, not a sequence. The two concepts are closely related, but they're not the same. – Barry Cipra Mar 21 '21 at 16:10
  • I wouldn't call it "easier" than ratio test, but certainly an elementary approach is to compute the partial sums and then take the limit. – Theo Bendit Mar 21 '21 at 16:14
  • If you look at the most frequent sequences-and-series question you will learn how to compute the sum of this series exactly. – Hans Lundmark Mar 21 '21 at 21:24

2 Answers2

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1)Root test:

https://en.m.wikipedia.org/wiki/Root_test

2)Recall:

$e^n=$

$1+n+n^2/2!+n^3/3!+... > n^3/3!$

$\dfrac{n}{3^n} <\dfrac{n}{e^n} <\dfrac{3!}{n^2};$

$\sum \dfrac {1}{n^2}$ converges (comparison test).

Peter Szilas
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Hint If your series got strictly positive terms if

$$ \frac{u_{n+1}}{u_n} \underset{n \rightarrow +\infty}{\rightarrow} \ell<1 $$ then $\sum u_n $ converges

Atmos
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