Loser66
  • Loser66
Explain me, please.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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dan815
  • dan815
for a decreasing sequence bn that summation converges
dan815
  • dan815
is it dirichlet?
Loser66
  • Loser66
Those are what my prof did in class.
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Loser66
  • Loser66
The red parts are my explanation on the lecture so that I remember what is going on when reviewing for test. :)
dan815
  • dan815
it is still downloading, its going very slow
Loser66
  • Loser66
dan!!
dan815
  • dan815
ok hmm i see
dan815
  • dan815
lets take cos(ntheta)= Real part of {e^(itheta)}
dan815
  • dan815
what they are using as their argument is that hey look just like how an alternating sequence of (-1)^n multiplied by a decreasing series goes to 0, this also does the same thing
dan815
  • dan815
because for any thetea u pick, for varying n, you keep getting some number of positive and some number of negatives , both equal to each other, tho the addition of this decreasing bn sequences works slightly differently
dan815
  • dan815
but you can still definately show that you will be subtracting greater part than what u are adding in the next set
dan815
  • dan815
here is a picture to help you see
dan815
  • dan815
|dw:1441645035166:dw|
dan815
  • dan815
varying theta is just like varying the period you sample your dots
Loser66
  • Loser66
next?
dan815
  • dan815
|dw:1441645239534:dw|
dan815
  • dan815
so in fact this is not much different from alternating series at all, its just put in a different way
dan815
  • dan815
let f(n) be some decreasing function for the first 3 values of n let cosntheta be positive for the next 3 values of n let cos n*theta be negative can you see why now, this has to be convergin
dan815
  • dan815
liket lets take a random decreasing function
dan815
  • dan815
|dw:1441645504019:dw|
dan815
  • dan815
|dw:1441645527838:dw|
Loser66
  • Loser66
|dw:1441645615856:dw|
dan815
  • dan815
thats wht u are doing with chos (ntheta) you are taking samples in each part of that curve adding and subtracting
Loser66
  • Loser66
dan, slow down on let f(n) be some decreasing function for the first 3 values of n let cosntheta be positive for the next 3 values of n let cos n*theta be negative can you see why now, this has to be convergin
anonymous
  • anonymous
wassup dan
dan815
  • dan815
|dw:1441645682902:dw|
Loser66
  • Loser66
f(n) = cos (ntheta) 3 values of n let cos n theta > 0
dan815
  • dan815
these could be your sambled points when you multiply by cos n theta
dan815
  • dan815
yeah like
dan815
  • dan815
|dw:1441645810574:dw|
Loser66
  • Loser66
|dw:1441645965886:dw|
Loser66
  • Loser66
They are the value of cos (n theta) when n = 1, 2, 3... right?
dan815
  • dan815
|dw:1441646040644:dw|
Loser66
  • Loser66
yes
dan815
  • dan815
yeah
dan815
  • dan815
now you say the sum of all these samples values in between is bounded by that formula they gave u up at the top
Loser66
  • Loser66
suppose 0
Loser66
  • Loser66
now what?
Loser66
  • Loser66
sum cos (n theta) = cos (0theta) + cos (1 theta) + cos (2 theta) + ..... cos (ntheta) , right? then?
Loser66
  • Loser66
is it not that it is =0?
dan815
  • dan815
|dw:1441646218632:dw|
dan815
  • dan815
now u can use that bound
anonymous
  • anonymous
@Adamlevine1986
dan815
  • dan815
notice how if theta -->0 its going to finity as u will be sampling an infinite points in each of the interval
Loser66
  • Loser66
then?
dan815
  • dan815
@empty how do u show this series converges
Loser66
  • Loser66
Is it not that if it is bounded and continuous, it converges?
dan815
  • dan815
its kind of weird to think about actually
dan815
  • dan815
like say you take an finite number of points on the highest parto of your decreasing function
dan815
  • dan815
|dw:1441646971233:dw|
dan815
  • dan815
now lets say u subtract an infinite set of points but lesser value.. the difference between these 2 is still infinity is it notq
dan815
  • dan815
|dw:1441647101921:dw|
dan815
  • dan815
but then u subtraced with some other difference an infinite set of points when compared to the next block of addition
Loser66
  • Loser66
I gotta go. You guys PPPPPPLLLLLEASE, freely discuss. I' ll take it later. Thanks in advance.

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