Reimann sum

gambix

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Jun 24, 2015
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Hello ,
I have the function f : [ 0 , pi/2] -> R , f(x) = cos(x) and i must find the limit when n goes to infinity of
cef29b5cb820e2e6837eeabdd06d3f60.png

Now, if
ab5a68823e26c105c0036d80f3fea988.png
were something like
878bd532f1718635c637124be801e4d9.png
then the limit would be 1 due to fact that integral from x=0 to x=1 of cos(x) = 1 , but i must prove or at least solve in some way the limit.
Any help will be much appreciated !
 
Hello ,
I have the function f : [ 0 , pi/2] -> R , f(x) = cos(x) and i must find the limit when n goes to infinity of
cef29b5cb820e2e6837eeabdd06d3f60.png

Now, if
ab5a68823e26c105c0036d80f3fea988.png
were something like
878bd532f1718635c637124be801e4d9.png
then the limit would be 1 due to fact that integral from x=0 to x=1 of cos(x) = 1 , but i must prove or at least solve in some way the limit.
I will tell you that post is so much nonsense. You start with \(\displaystyle \left[0,\frac{\pi}{2}\right]~\&~f(x)=\cos(x)\)
to \(\displaystyle \left[0,1\right]~\&~f(x)=\cos(x)\) which is it?
Before you answer consider: \(\displaystyle \int_0^1 {\cos (t)dt} \ne \int_0^{\frac{\pi }{2}} {\cos (t)dt} = 1\)
In other words, \(\displaystyle \int_0^1 {\cos (t)dt} \ne 1\) as you seem to think.

Also, that sum you posted has absolutely nothing to do with a Reimann Sum for this integral.
For example: where on earth did you get \(\displaystyle \frac{1}{\sqrt{n}}~?\)
 
Hello ,
I have the function f : [ 0 , pi/2] -> R , f(x) = cos(x) and i must find the limit when n goes to infinity of
cef29b5cb820e2e6837eeabdd06d3f60.png

Now, if
ab5a68823e26c105c0036d80f3fea988.png
were something like
878bd532f1718635c637124be801e4d9.png
then the limit would be 1 due to fact that integral from x=0 to x=1 of cos(x) = 1 , but i must prove or at least solve in some way the limit.
Any help will be much appreciated !

What are you allowed to use for cos(x)? For example, are you allowed to use
1-cos(x) = 1 - (\(\displaystyle 1\, -\, \frac{1}{2}x^2\, +\, \frac{1}{4!}x^4\,
-\, \frac{1}{6!}x^6\, ...\))
and use the alternating series bounds and/or that cos(x) is monotonic on the given interval?

Is the sequence increasing and bounded above?
 
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