Subscribe via feed.

## A complex sum

Posted by Chris on November 20, 2011 – 1:58 pm

Evaluate Sum(n = 0 to ∞, cos(n x)/2^n), where cos(x) = 1/5.

Indirect hint, really an excuse to point out the remarkable proof of a distant cousin to the above problem, see: http://en.wikipedia.org/wiki/Gaussian_integral

This post is under “Mathemagic, MathsChallenge” and has 5 respond so far.
If you enjoy this article, make sure you subscribe to my RSS Feed.

### 5 Responds so far- Add one»

1. 1. Carc Said：

7/6?

2. 2. Chris Said：

Hi Carc. I’m going to assume you made a “typo” somewhere: it’s 6/7.

I’ve just skimmed the link I gave. Sadly my memory of the proof was faulty. So, keeping my hint very indirect, see http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28III.29_.E2.80.93_trigonometric_integrals (I used to do that sort of stuff 40 years ago – it’s much easier than it looks).

3. 3. slavy Said：

Hi Chris. I think the hint is slightly misleading. For those who are working on the problem: Don’t try to come up with some fancy integral of a real-valued function and solve it as a (part of a) contour integral of a complex function! The only thing you need is actually this link http://en.wikipedia.org/wiki/Euler%27s_formula

4. 4. Chris Said：

Hi Slavy. I probably was being too indirect. All I was really trying to avoid saying is, “use Euler’s formula”. I’m dying to know if that’s what Carc did.

5. 5. Chris Said：

Letting “Re” be “the real part of”, then cos(nx) = Re(einx)
So the original sum is Re(Sum(n = 0 to ∞, einx / 2n))
= Re(Sum(n = 0 to ∞, (eix/2)n))

But 1 + z + z2 + z3 + … = 1/(1 – z), for |z| < 1.
So Sum … = 1/(1 – eix/2) = 1/(1 – (cos(x) + i sin(x))/2).
We are given that cos(x) = 1/5 => sin(x) = ±(2/5)√6
(I used sin2(x) + cos2(x) = 1, to get that)
=> Sum … = 1/(1 – ((1/5) ± i (2/5)√6)/2)
=> Sum… = 10/(10 -(1 ± 2 i √6)) = 10/(9 -± 2 i √6))

Using 1/(x + iy) = (x – iy)/((x + iy)(x – iy)) = (x – iy)/(x2 + y2)
=> Sum… = (90 ± 20 i √6)/105. The real part is 90/105 = 6/7.

PHP Warning: PHP Startup: Unable to load dynamic library 'C:\Program Files (x86)\Parallels\Plesk\Additional\PleskPHP5\ext\php_mssql.dll' - The specified module could not be found. in Unknown on line 0 PHP Warning: PHP Startup: Unable to load dynamic library 'C:\Program Files (x86)\Parallels\Plesk\Additional\PleskPHP5\ext\php_pdo_mssql.dll' - The specified module could not be found. in Unknown on line 0