Proof infinite series $1 + 2 + 3 +\cdots +n+\cdots$ diverges

Question

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.

Answer 1

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 $n\in \Bbb N:$

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

Answer 2

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. ;)

Answer 3

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 $n\rightarrow\infty$.