Math Is Fun Forum

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Real Analysis Help

Last edited by sumpm1 (2009-04-28 13:40:22)

jonathan s

Guest

Re: Real Analysis Help

1)The sequence obviously converges and the limit depends on the value of a. For example, if you let a = 1, the limit is 0.5.   Try showing that it is Cauchy and hence convergent.

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Real Analysis Help

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Real Analysis Help

I think that if you prove that the limit equals a/2 with a being a positive number you will have proven that the sequence is bounded and convergent.

http://www.sosmath.com/calculus/sequence/limit/limit.html

Last edited by bobbym (2009-04-29 02:38:17)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Real Analysis Help

Heres how to find the limit:

Now make the substitutions

Plug them in (A). This is the new limit.

Now since (B) is of the form 0/0 we can differentiate the numerator and the denominator with respect to u. This leaves:

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

whatismath

Member

Registered: 2007-04-10

Posts: 19

Re: Real Analysis Help

This is basically the (algebraic) technique of
rationalizing the numerator: multiply both
numerator and denominator by

and after
simplifying the numerator, divide both
numerator and denominator by

.
Take limit.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Real Analysis Help

Hi whatismath;

  I tried that idea and could not make it work. The indeterminate forms produced had to be differentiated and did not get simpler. Can you demonstrate your method as I would like to see a shorter method than mine.

Last edited by bobbym (2009-05-10 22:29:54)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

whatismath

Member

Registered: 2007-04-10

Posts: 19

Re: Real Analysis Help

Sorry, we should multiply

to both numerator and denominator:

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Real Analysis Help

Hi whatismath;

Indeed it is shorter, and doesn't require a substitution like mine does. Thanks for providing it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.