# Tail of a Sequence

• Aug 13th 2011, 08:47 AM
veronicak5678
Tail of a Sequence
Prove a sequence S_n converges iff its tail S_n+M converges.
• Aug 13th 2011, 10:10 AM
abhishekkgp
Re: Tail of a Sequence
Quote:

Originally Posted by veronicak5678
Prove a sequence S_n converges iff its tail S_n+M converges.

do you mean $S_n$ converges iff $S_{n+M}$ converges?
• Aug 13th 2011, 10:45 AM
veronicak5678
Re: Tail of a Sequence
Yes, that's what I mean. Sorry I don't know Latex!
So... any ideas?
• Aug 13th 2011, 07:51 PM
abhishekkgp
Re: Tail of a Sequence
Quote:

Originally Posted by veronicak5678
Yes, that's what I mean. Sorry I don't know Latex!
So... any ideas?

you just have to use the definition of convergence.

Assume $S_n \rightarrow a$. we will show that $S_{n+M} \rightarrow a$.

$S_n \rightarrow a$ gives us:
$\forall \epsilon >0, \exists N \text{ such that } n>N \Rightarrow |S_n-a|<\epsilon$

To prove $S_{n+M} \rightarrow a$, that is:

$\forall \epsilon>0, \exists N^{'} \text{ such that } n>N^{'} \Rightarrow |S_{n+M}-a|<\epsilon$. We easily see that $N^{'}=N-M$ does the job.

Now can you try and do the 'only if' part.
• Aug 13th 2011, 09:45 PM
veronicak5678
Re: Tail of a Sequence
I see what I was doing wrong. Thanks everyone.