How can I calculate it for by using the fact that ?
I tried letting u = cos(x), then
But I don't think this is right because the first term should be 0.5, not 1... and I don't see how a -0.5 might pop out of this series of cos terms....and even if it somehow does, I think this question is not meant to be that difficult...
Any ideas?
Thanks
I tried letting u = cos(x), then
But I don't think this is right because the first term should be 0.5, not 1... and I don't see how a -0.5 might pop out of this series of cos terms....and even if it somehow does, I think this question is not meant to be that difficult...
The answer I am trying to work towards is
Well, of course that 1 - cos^2(x) + cos^4(x)-.... cannot be a valid answer since you're not giving a McClaurin series IN X but in cos x, and that's not, apparently, what you were asked to do.
Sometimes , in some maths forum, there had been discussions as whether the infinite series 1 -1 + 1 - 1 +.... """converges""" to 1/2, and one of the reasons is precisely this developement: if you use the usual McClaurin series for cos(x), you get:
== cos x = 1 - x^2/2 + x^4/4 -....
== cox^2(x) = (1 - x^2/2 +...)^2 = 1 - x^2 +...
== cox^4(x) = (1 - x^2/2 +...)^4 = 1 - 2x^2 +....
etc.
Now, if you want to know the value of f(x) = 1/[1 + cox^2(x)] at x = 0, which clearly is 1/2, using the above result with
1/[1 + cosx^2(x)] = 1 - cos^2(x) + cos^4(x) -...
and you look at the constant coefficient in the RH, you get 1 - 1 + 1 - 1 +..., so "it must be" that its sum is 1/2.
You can read about this Grandi's Series here Grandi's series - Wikipedia, the free encyclopedia
and if you know something about Cesaro's sums then it may be make a little more sense.
Tonio