# Thread: Prove the tail of a sequence converges?

1. ## Prove the tail of a sequence converges?

The problem is prove that {Sn} converges if and only if {Tn} converges where {Tn} = {Sn+M) for n=1,2,3,... and M is some integer (perhaps large).

I know to start out going to the right in direction you assume {Sn} converges. And since it converges for all epsilon>0 there exists an N1 in Natural Numbers such that |Sn1 - L| < epsilon for all n1 greater than or equal to N1.
I know I want to show that for all epsilon>0 there exists an N2 in natural numbers such that |Tn2 - P| < epsilon for all n2 greater than or equal to N2.

I've thought of several ways to approach it. Since it has all the same numbers, the sequences are equal, Tn just reaches the later values of the sequence much quicker. However I do not know how to use that correctly in a proof.

Any help is very much appreciated!!

2. Originally Posted by mathgirl13
The problem is prove that {Sn} converges if and only if {Tn} converges where {Tn} = {Sn+M) for n=1,2,3,... and M is some integer (perhaps large).
You mean " $T_n= S_{n+M}$". I first thought you meant " $T_n= S_n+ M$ and got really confused!

I know to start out going to the right in direction you assume {Sn} converges. And since it converges for all epsilon>0 there exists an N1 in Natural Numbers such that |Sn1 - L| < epsilon for all n1 greater than or equal to N1.
I know I want to show that for all epsilon>0 there exists an N2 in natural numbers such that |Tn2 - P| < epsilon for all n2 greater than or equal to N2.

I've thought of several ways to approach it. Since it has all the same numbers, the sequences are equal, Tn just reaches the later values of the sequence much quicker. However I do not know how to use that correctly in a proof.

Any help is very much appreciated!!
Yes, given any $\epsilon> 0$, there exist N such that if n> N, $|S_n- L|< \epsilon$.

Now look at $|T_n- L|= |S_{n+M}- L|$. That will be less than $\epsilon$ as long as n+M> N or n> N+ M. That's all you need to do for that direction.

Since this is "if and only if" you also need to prove that if $\{T_n\}$ converges, so does $\{S_n\}$. Do that in exactly the same way.

3. Awesome!! Thank you so much!!

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# proof that if tail end of a series

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