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Convergence of a Sequence 
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#1
Oct1505, 03:28 PM

P: n/a

"Let [tex]k\in \mathbb{N}[/tex] and [tex]a_0=k[/tex]. Let [tex]a_n=\sqrt{k+a_{n1}}, \forall n\geq1[/tex] Prove that [tex]a_n[/tex] converges."
If we look at the similar sequence b_0 = k and b_n = sqrt(a_n1), then that sequence obviously converges to 1. Unfortunately, b_n<a_n so I can't use the squeeze theorem. Any hints would be nice. 


#2
Oct1505, 04:03 PM

P: 42

I would say let lim an = s also lim an1 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



#3
Oct1605, 12:23 AM

P: 347

Have you tried checking if a_n is monotonic & bounded? 


#4
Oct1605, 12:39 AM

HW Helper
PF Gold
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. 


#5
Oct1605, 09:54 AM

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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