Prove that if a sum converges, its nth term is 0.

**davismj** · January 26th 2010, 03:49 PM

I apologize, I'm so new at proofs. This should be very easy, but I'm not sure how to say it. This is what I've come up with. Is it sufficient? Is there another way I should be looking at this?

Thanks in advance!

**Krizalid** · January 26th 2010, 03:58 PM

I don't usually prove statements by using contradictions, but here's how I'd solve this:

Given $\sum_{n=1}^\infty a_n<\infty,$ then $\lim_{n\to\infty}\sum_{k=1}^n a_k$ exists, and so does $\lim_{n\to\infty}\sum_{k=1}^{n-1} a_k,$ hence $\underset{n\to \infty }{\mathop{\lim }}\,\left( \sum\limits_{k=1}^{n}{a_{k}}-\sum\limits_{k=1}^{n-1}{a_{k}} \right)=\underset{n\to \infty }{\mathop{\lim }}\,\left( \sum\limits_{k=1}^{n-1}{a_{k}}+a_{n}-\sum\limits_{k=1}^{n-1}{a_{k}} \right)=0,$ then $\underset{n\to \infty }{\mathop{\lim }}\,a_{n}=0,$ as claimed.

**Drexel28** · January 26th 2010, 05:59 PM

Originally Posted by davismj

I apologize, I'm so new at proofs. This should be very easy, but I'm not sure how to say it. This is what I've come up with. Is it sufficient? Is there another way I should be looking at this?

Thanks in advance!

Let $\sigma_m=\sum_{n=1}^{m}a_n$ . Since, $\lim\text{ }\sigma_m$ exists (i.e. it converges) we have that it is Cauchy. Thus, for every $\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that $N<\ell\leqslant k\implies\left|\sigma_k-\sigma_\ell\right|=\left|\sum_{n=1}^{\ell}-\sum_{n=1}^{k}\right|=\left|\sum_{n=\ell}^{k}a_n\r ight|<\varepsilon$ . Taking $\ell=k$ we see that for every $\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that $N<\ell\implies \left|\sum_{n=\ell}^{\ell}a_n\right|=\left|a_\ell\ right|<\varepsilon$ . Therefore, $a_\ell\to0$ .

**davismj** · January 26th 2010, 06:29 PM

Originally Posted by Drexel28

Let $\sigma_m=\sum_{n=1}^{m}a_n$ . Since, $\lim\text{ }\sigma_m$ exists (i.e. it converges) we have that it is Cauchy. Thus, for every $\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that $N<\ell\leqslant k\implies\left|\sigma_k-\sigma_\ell\right|=\left|\sum_{n=1}^{\ell}-\sum_{n=1}^{k}\right|=\left|\sum_{n=\ell}^{k}a_n\r ight|<\varepsilon$ . Taking $\ell=k$ we see that for every $\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that $N<\ell\implies \left|\sum_{n=\ell}^{\ell}a_n\right|=\left|a_\ell\ right|<\varepsilon$ . Therefore, $a_\ell\to0$ .

This is what I was trying to say, but I had no idea how to say it. I kept writing things and erasing it because it didn't make sense like I wanted it to

Thank you so much!

Thread: Prove that if a sum converges, its nth term is 0.

LinkBack

Thread Tools

Display

Prove that if a sum converges, its nth term is 0.

Similar Math Help Forum Discussions

Prove this series converges

converges to zero prove

prove that 1/(2^(2n)) converges

Prove that this converges

Prove that the series 1/n^2 converges

Search Tags