Does the Tail of a Convergent Series Also Converge to Zero?

[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

{\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

 

[John Park]

Can you write Σ₁^∞a_n - Σ_N+1^∞a_n as a sum?

It might be helpful to put Σ₁^∞a_n = c .

 

[Mark44]

{\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?

 

[mathman]

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