Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
Solve summation involving trig?
Evaluate sum from n=0 to infinity of cos(nx)/(2^n) if cos(x)=4/5.
1 Answer
- 1 decade agoFavorite Answer
Now cos nx = the real part of exp(i n x), so the sum is the real part of summation of n=0 to infinity of exp( i n x)/2^n. This is a geometric series with first term a = 1 and ratio r = exp(i x)/2. So the infinite sum is a/(1-r) which equals 1/(1 - exp(i x)/2) = 2/(2-exp(ix)). The real part of this is the answer to the question. Now
2/(2 - exp(ix)) = 2/(2-cosx - i sinx) = 2(2-cosx + i sinx)/[(2-cosx)^2 + (sinx)^2]. So the real part is
2(2-cosx)//[(2-cosx)^2 + (sinx(^2)] which can be simplified to 2(2-cosx)/(5-2cosx).
