## Loser66 one year ago Explain me, please.

1. dan815

for a decreasing sequence bn that summation converges

2. dan815

is it dirichlet?

3. Loser66

Those are what my prof did in class.

4. Loser66

The red parts are my explanation on the lecture so that I remember what is going on when reviewing for test. :)

5. dan815

6. Loser66

dan!!

7. dan815

ok hmm i see

8. dan815

lets take cos(ntheta)= Real part of {e^(itheta)}

9. 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

10. 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

11. dan815

but you can still definately show that you will be subtracting greater part than what u are adding in the next set

12. dan815

13. dan815

|dw:1441645035166:dw|

14. dan815

varying theta is just like varying the period you sample your dots

15. Loser66

next?

16. dan815

|dw:1441645239534:dw|

17. dan815

so in fact this is not much different from alternating series at all, its just put in a different way

18. 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

19. dan815

liket lets take a random decreasing function

20. dan815

|dw:1441645504019:dw|

21. dan815

|dw:1441645527838:dw|

22. Loser66

|dw:1441645615856:dw|

23. dan815

thats wht u are doing with chos (ntheta) you are taking samples in each part of that curve adding and subtracting

24. 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

25. anonymous

wassup dan

26. dan815

|dw:1441645682902:dw|

27. Loser66

f(n) = cos (ntheta) 3 values of n let cos n theta > 0

28. dan815

these could be your sambled points when you multiply by cos n theta

29. dan815

yeah like

30. dan815

|dw:1441645810574:dw|

31. Loser66

|dw:1441645965886:dw|

32. Loser66

They are the value of cos (n theta) when n = 1, 2, 3... right?

33. dan815

|dw:1441646040644:dw|

34. Loser66

yes

35. dan815

yeah

36. 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

37. Loser66

suppose 0<theta < pi, then it limits on |dw:1441646174573:dw|

38. Loser66

now what?

39. Loser66

sum cos (n theta) = cos (0theta) + cos (1 theta) + cos (2 theta) + ..... cos (ntheta) , right? then?

40. Loser66

is it not that it is =0?

41. dan815

|dw:1441646218632:dw|

42. dan815

now u can use that bound

43. anonymous

44. dan815

notice how if theta -->0 its going to finity as u will be sampling an infinite points in each of the interval

45. Loser66

then?

46. dan815

@empty how do u show this series converges

47. Loser66

Is it not that if it is bounded and continuous, it converges?

48. dan815

its kind of weird to think about actually

49. dan815

like say you take an finite number of points on the highest parto of your decreasing function

50. dan815

|dw:1441646971233:dw|

51. 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

52. dan815

|dw:1441647101921:dw|

53. dan815

but then u subtraced with some other difference an infinite set of points when compared to the next block of addition

54. Loser66

I gotta go. You guys PPPPPPLLLLLEASE, freely discuss. I' ll take it later. Thanks in advance.