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Montone/Dominated convergence

I'm just starting out learning about monotone and dominated convergence theorems and I'm struggling to really get my head around the concepts.

Is MON the idea of finding a sequence which is increasing as n gets bigger and then finding the limit of the integral as n goes to infinity?
And DOM the idea of finding a function g(x) which is bigger than the function we are integrating for every value of x?

I've got a couple of questions here that require the use of both techniques:

Use both Dominated and Montone convergence to find limit as n goes to infinity of

a)

\displaystyle\int^\frac{\pi}{2}_0 \frac{\cos(x)}{1+\frac{x^3}{n}}\ dx

\displaystyle\int^\pi_0 \frac{\cos(x)}{1+\frac{x^3}{n}}\ dx

for a) using MON, can I simply say that as n increases, our function is increasing and as n get to infinity, the function is equal to cos(x)
and then

\displaystyle\int^\frac{\pi}{2}_0 \cos(x)\ dx = 0

?

Using DOM, I want to try and find a function whose integral is finite between pi/2 and 0 and which is bigger than

\displaystyle\frac{\cos(x)}{1+ \frac{x^3}{n}}\

, but I really am not sure what to use?

Also I'm not sure really how part b) differs apart from the change of limits?
Not sure what effect this will have!

Any help would be massively appreciated!
Thank you!

Reply 1

DFranklin

I think the phrase "converges pointwise" is important here; the point of the Lebesgue convergence theorems is that the limit of the integrals f_n is the same as the integral of the pointwise convergent limit f_n. In general phrases like "as n gets to infinity, the function is equal to cos(x)" are dangerous since on one side your talking about the function globally and on the other side you're talking about the value at a point.

For the examples given, convergence is uniform, so this is a bit moot, but I'd still probably dock you a mark or so for being unclear about it.

Note also that

\int_0^{\pi/2} \cos(x)\,dx

does NOT equal 0.

For the DOM part, I think you need to look more closely at the statement of the theorem. You might also want to think about why you can't simply use the same argument as (a).

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Montone/Dominated convergence

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