# prove that a sequence converges

• Apr 30th 2010, 03:18 PM
santiagos11
prove that a sequence converges
Prove that the sequence $x_{n}=n^{1/n}$ converges.
I know it converges to 1. We need to use the definition of sequence convergence: $x_{n} \rightarrow L \Leftrightarrow(\forall\epsilon>0)(\exists N\in \mathbb{N})(\forall n\in \mathbb{N})(n>N \Rightarrow |x_{n}-L|<\epsilon)$
Another thing, obviously $|x_{n}-L|=|n^{1/n}-1|=n^{1/n}-1$ since $n^{1/n}-1\geq0$
• Apr 30th 2010, 03:58 PM
Plato
Quote:

Originally Posted by santiagos11
Prove that the sequence x(n)=n^(1/n) converges.

Let $y_n=\sqrt[n]{n} - 1$ so $y_n\ge 0$ and
by the binominal theorem $n=(y_n+1)^n>\frac{n(n-1)}{2}y_n^2$.
Therefore $0\le y_n<\sqrt{\frac{2}{n-1}}$.
• Apr 30th 2010, 04:36 PM
santiagos11
Well, you are certanly right. But how does that fit the standard definition?
• Apr 30th 2010, 05:02 PM
Plato
Quote:

Originally Posted by santiagos11
Well, you are certanly right. But how does that fit the standard definition?

Having taught this 'stuf' for thirty+ years I have never seen a $\epsilon\-\delta$ proof given for this limit.
So I can't imagine what you mean be 'standard'.
• Apr 30th 2010, 06:04 PM
santiagos11
For example, here is a partial proof:
Let $\epsilon>0$. Clearly $1\leq n^{1/n}\leq 2 \ \forall n\in \mathbb{N}$
case 1: $\epsilon>1$
Let $N=1$. Suppose $n>N$. Then $|x_{n}-1|=|n^{1/n}-1|=n^{1/n}-1$. On the other hand $\epsilon + 1 > 2 \geq n^{1/n}$, so $n^{1/n}-1<\epsilon$. Therefore, $|x_{n}-1|<\epsilon$.
Do you see how this follows the definition that I gave in the top?
Now it remains to show the same, in the case $0<\epsilon\leq1$
• Apr 30th 2010, 07:15 PM
Plato
Quote:

Originally Posted by santiagos11
For example, here is a partial proof:
Let $\epsilon>0$. Clearly $1\leq n^{1/n}\leq 2 \ \forall n\in \mathbb{N}$
case 1: $\epsilon>1$
Let $N=1$. Suppose $n>N$. Then $|x_{n}-1|=|n^{1/n}-1|=n^{1/n}-1$. On the other hand $\epsilon + 1 > 2 \geq n^{1/n}$, so $n^{1/n}-1<\epsilon$. Therefore, $|x_{n}-1|<\epsilon$.
Do you see how this follows the definition that I gave in the top?
Now it remains to show the same, in the case $0<\epsilon\leq1$

That is a waste of effort. There is absolutely no reason to reinvent the wheel.
The object of proof at this level is to deepen understanding, not to show off.
• May 1st 2010, 07:48 PM
santiagos11
Well yeach I gues you are right. However I do have another problem:
Prove that that the following sequence,defined recursively, converges or diverges:
$x_{1}=1$
$x_{n}=\sqrt{1+x_{n-1}}$