Proving Convergence of Sequences at Infinity: A Case Study in Advanced Calculus

[WHOAguitarninja]

Homework Statement

Discuss the convergence of the following sequences at infinity.
a)\sqrt{n+1}-\sqrt{n}
b)\sqrt{n}(\sqrt{n+1}-\sqrt{n})
c)n(\sqrt{n+1}-\sqrt{n})I've already solved a, and if I can solve b then I have c automatically.

This is for an undergraduate advanced calc course so all we've really covered is basic proving techniques, so I can't use derivatives or any sort of route/ratio test type thing.

I know the series converges to 1/2, so what I've done so far has been mostly aimed at trying to find another sequence that converges to a non 0 real number such that when I multiply or divide the above sequence (b) by it I get something easier to work with. This has proved unsuccesful. From examination and playing around it seems the sequence is monotonicly increasing, though I'm not entirely sure how to prove that without taking derivatives. I tried to directly show a(n+1)-a(n) is always positive where a(n) is the nth entry in the sequence. Still grinding that out but it's not going well, I don't think I'll get anything concrete.

Like I said...b is the key. If I can get b I can get c.

EDIT - Ok I just figured out a way to show it's monotonic. Now I just need to show it's bounded and it's lub is 1/2. Which is what I've been stuck on all day.

 

[Dick]

Multiply your sequences by (sqrt(n+1)+sqrt(n))/(sqrt(n+1)+sqrt(n)) and do some algebra.

 

[WHOAguitarninja]

Multiply your sequences by (sqrt(n+1)+sqrt(n))/(sqrt(n+1)+sqrt(n)) and do some algebra.

You know...it's funny...I had considered multiplying by sqrt(n+1)+sqrt)n), but decided against it since it limits to infinity and I figured it wouldn't tell me anything...didn't even consider just multiplying what you suggested.

Thanks a million.