# I Proof of series`s tail limit

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1. Mar 16, 2017

### DaniV

{\displaystyle \sum_{n=1}^{\infty}a_{n}}
is converage, For N\in
\mathbb{N}

\sum_{n=N+1}^{\infty}an
is also converage

proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0
Code (Text):
{\displaystyle \sum_{n=1}^{\infty}a_{n}}
is converage, For N\in
\mathbb{N}

\sum_{n=N+1}^{\infty}an
is also converage

proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0

Last edited: Mar 16, 2017
2. Mar 16, 2017

### John Park

Can you write Σ1an - ΣN+1an as a sum?

It might be helpful to put Σ1an = c .

3. Mar 19, 2017

### Staff: Mentor

Please take a look at our LaTeX tutorial -- https://www.physicsforums.com/help/latexhelp/
I believe this is what you were trying to convey:
$\sum_{n=1}^{\infty}a_{n}$ converges.

For $N \in \mathbb{N}, \sum_{n = N+1}^{\infty}a_n$ also converges.

What does it mean for a series to converge? How is this defined?

Also, is this a homework problem?

4. Mar 19, 2017

### mathman

The difference between a full series and a tail is finite, so convergence is te same for both.