Convergence of sequence

[mathmari]

Hi! I am looking at an exercise in Analysis, which ask me to find the limit of the sequence \(\displaystyle x_{n+1}=x_{n}+\frac{1}{x_{n}^2} \) .

Firstly, to show that this sequence converges, I have to show that the sequence is monotonic, in this case \(\displaystyle x_{n} \) is increasing, then I have to show that the sequence is bounded above. How can I show this?

Having shown the above, let \(\displaystyle x \) be the limit of the sequence, then \(\displaystyle x=x+\frac{1}{x^2} \), that means \(\displaystyle \frac{1}{x^2}=0 \). Isn't this wrong?

 

[HallsofIvy]

First, you can see that \(\displaystyle x_{n+1}\) is \(\displaystyle x_n\) plus a positive number. That tells you it is increasing.
But you cannot show that the sequence is bounded- it isn't. That is what your "\(\displaystyle \frac{1}{n}= 0\)" shows. If it were bounded it would be convergent but you have shown that it cannot converge to any finite number. The answer to this exercise is that the sequence does NOT converge.

 

[mathmari]

Ok!! Thank you very much!!