Proof of series`s tail limit

  • #1
34
3
{\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:
{\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:

Answers and Replies

  • #2
193
29
Can you write Σ1an - ΣN+1an as a sum?

It might be helpful to put Σ1an = c .
 
  • #3
35,428
7,288
{\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
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
mathman
Science Advisor
8,008
518
The difference between a full series and a tail is finite, so convergence is te same for both.
 

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