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Question:

By considering the partial sums for S, that is

$S_n = 1 + 2 + 3 +\cdots +n$

show that the infinite series S does not converge.

My answer :

I tried to attempt this question, but I was able to prove that the series converges to $-1/12$ instead of diverges. Can anyone help me to prove it diverges instead of converges? your help is appreciated.

Martin Sleziak
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Amanda Kim Hyuen
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  • What? It converges? – Hanul Jeon May 03 '15 at 05:20
  • I proof that it converges , but the question want me to show that it diverges . I need help for that .Do I use the proofing of harmonic series ? if not can you tell me how to solve this ? – Amanda Kim Hyuen May 03 '15 at 05:22
  • @ Solid Snake, I have not learn Cesàro summation , can you show me how you do it ? Cesaro summation is for the case of 1-1+1-1+1..... , but this is not the case are you sure about using that to solve this? – Amanda Kim Hyuen May 03 '15 at 05:24
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    Direct calculation shows $1+2+\cdots + n = n(n+1)/2$. Can you calculate the limit $\lim_{n\to\infty} n(n+1)/2$? – Hanul Jeon May 03 '15 at 05:25
  • this "converges" to $-\frac{1}{12}$ using an extension of $\zeta$ function. i don't know much about this but your serie do not converges – L F May 03 '15 at 05:26
  • @tetori , if lim n→∞n(n+1)/2 diverges , the infinite series S diverges too ? is that correct ? – Amanda Kim Hyuen May 03 '15 at 05:28
  • Yes, remind the definition of infinite series - infinite series $\sum_n a_n$ defined as the limit of finite series $\lim_N\sum_{n<N} a_n$. – Hanul Jeon May 03 '15 at 05:31
  • I think we should say that this series diverge, but with black magic the sum is $-1/12$. You can find many questions about this on MSE (probably on MO as well, I'm not sure). – MonkeyKing May 03 '15 at 05:39

3 Answers3

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The argument you're using to prove it "converges" to $-1/12$ is using a different idea of summability, here, you're being asked to prove it diverges in the common sense: you must prove the sequence $\{S_n\}_n$ diverges.

This is an easy task since for all $n\in \Bbb N:$

$$|S_n|=S_n=1+2+\cdots +n\geq n\ \ .$$

Daniel
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  • My answer:

    S0 = 1 S1 = 1 + 2 = 3 S2 = 1 + 2 + 4 = 7 S3 = 1 + 2 + 4 + 8 = 15

    So we see that Sn → ∞ meaning that the sum diverges.

    Please forgive me for my incorrect way of commenting in this website . Does my answer utilize your explanation ? Is it correct ?

     – Amanda Kim Hyuen May 03 '15 at 05:35
  • Your argument is fine, lacks of rigurosity but it shows what's going on. A more formal proof would go as I showed in my answer. – Daniel May 03 '15 at 05:38
  • Ok thx for the big help , appreciated . :D – Amanda Kim Hyuen May 03 '15 at 05:38
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For a serie to convergence, one must have $$\mid s_{n+1}-s_n\mid\rightarrow 0$$ but in this case this is equivalent to $n\rightarrow 0$. Clearly, this is not the case as $n\rightarrow\infty$.

Jean-François Gagnon
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A Serie diverges if for all $A\in\mathbb{N}$, there exists some $n_0\in\mathbb{N}$ such $S_{n_0}>A$

set $n_0=A$.

also, if it is convergent, then it should be bounded. ;)

L F
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