Register to reply

Limits, n, n+1

by Tclack
Tags: infinity, limit
Share this thread:
Aug17-10, 11:42 PM
P: 37
So I came across the statement:

Since [itex] x_n -> \inf [/itex]
then [itex] x_n_+_1 -> \inf [/itex]

This is very basic, But I'm already into recursive formulas for infinite series, so I should know why this is true. Does anyone have a small proof. An informal one will do.
Phys.Org News Partner Science news on
FIXD tells car drivers via smartphone what is wrong
Team pioneers strategy for creating new materials
Team defines new biodiversity metric
Aug18-10, 07:43 AM
Sci Advisor
HW Helper
P: 4,300
This is about as informal as they get, but ... if you delete from the second limit your first element, you shift everything down by one and you get the same limit again.
Aug18-10, 08:06 AM
P: 37
I thought of something.
As X_n goes to infinity, it passes X_(n+1)

Aug18-10, 08:14 AM
Sci Advisor
PF Gold
P: 39,552
Limits, n, n+1

I'm not sure what that means! For any n, n< n+1. But I think you mean that, for fixed N, eventually, n> N+1.

A little more precisely, if [itex]\{a_n\}[/itex] converges to A, then, for any [itex]\epsilon> 0[/itex], there exist N such that if n> N, [itex]|a_n- A|< \epsilon[/itex].

Now, if [itex]b_n= a_{n+1}[/itex], for any [itex]\epsilon> 0[/itex], take N'= N-1 where N is the number, above, for that same [itex]\epsilon[/itex].

Then if n> N' , n+1> N'+1= N so [itex]|b_n- A|= |a_{n+1}- A|< \epsilon[/itex], showing that [itex]\{b_n\}[/itex] also converges to A.

(Roughly speaking, [itex]\{a_n\}[/itex] and [itex]\{a_{n+1}\}[/itex] are really the same sequence, just with the "numbering" altered slighly. Of course, they have the same limit.)
Aug18-10, 08:22 AM
P: 3,014
You need to know what subsequences are:
You choose a strictly monotonically increasing sequences of natural numbers:

n_{k}, n_{k} \in \mathbb{N}, n_{k + 1} > n_{k}, \; k = 0, 1, \ldots

Then, a subsequence of the sequence [itex]\{x_{n}\}[/itex] is defined as:
\tilde{x}_{k} \equiv x_{n_{k}}

The (informal) theorem you will need to remember is:

Any subsequence has the same convergence properties and the same limit if convergent as its sequence.

How would you choose your subsequence?

Register to reply

Related Discussions
Limits, yes another limits thread. General Math 25
L'Hopitals rule and application in limits and limits Calculus & Beyond Homework 2
Limits problem help Calculus & Beyond Homework 2
Limits using basic analysis theorems and logic? Introductory Physics Homework 9
Sequences of positive numbers and limits Introductory Physics Homework 2