# Convergence of a Sequence

by Icebreaker
Tags: convergence, sequence
 P: n/a "Let $$k\in \mathbb{N}$$ and $$a_0=k$$. Let $$a_n=\sqrt{k+a_{n-1}}, \forall n\geq1$$ Prove that $$a_n$$ converges." If we look at the similar sequence b_0 = k and b_n = sqrt(a_n-1), then that sequence obviously converges to 1. Unfortunately, b_n
 P: 42 I would say let lim an = s also lim an-1 would still be s so you can use the limit properties and can get a quadratic with s^2 - s -k=0 you should be able to go from there
P: 347
 Quote by 1800bigk I would say let lim an = s also lim an-1 would still be s so you can use the limit properties and can get a quadratic with s^2 - s -k=0 you should be able to go from there
But you don't know if $\lim a_n$ exists.

Have you tried checking if a_n is monotonic & bounded?

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HW Helper
P: 1,198

## Convergence of a Sequence

You should first try to prove that the sequence is bounded.
Then if you show that it monotonically increases or decreases, you can prove that the sequence is convergent.
 P: n/a It can easily be shown that it's monotonically increasing. However, it's the bounded part that gets me. Maybe I can use Herschfeld's Convergence Theorem?

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