Cos x limiting sum (1 Viewer)

[shongaponga]

By using the graph of y = cos x for 0 ≤ x ≤ 2π, or otherwise, find those values of x satisfying the given domain for which the geometrical series 1 + 2 cos x + 4 cos²x + 8 cos³x + ... has a limiting sum.


I was wondering if anybody could please provide me a solution by using the 'or otherwise', I've done the question using the graph but just am not a fan of the method lol.

 

[Carrotsticks]

By using the graph of y = cos x for 0 ≤ x ≤ 2π, or otherwise, find those values of x satisfying the given domain for which the geometrical series 1 + 2 cos x + 4 cos²x + 8 cos³x + ... has a limiting sum.


I was wondering if anybody could please provide me a solution by using the 'or otherwise', I've done the question using the graph but just am not a fan of the method lol.

For a series to have a limiting sum, the ratio R must satisfy -1 < R < 1

In this case, the ratio is 2cos(x)

So -1 < 2cos(x) < 1

-1/2 < cos(x) < 1/2

Is this what you did? Then you used the graph of y=cos(x) to determine the bounds of the inequality?

 

[shongaponga]

Yes, my solution continued on from there.

cos x = 1/2 (0 ≤ x ≤ 2π)

when x = π/3; 2π - π/3 = 5π/3

cos x = -1/2

when x = π - π/3 = 2π/3, π + π/3, = 4π/3

.'. Series has a limiting sum for π/3 < x < 2π/3 and 4π/3 < x < 5π/3.

I recognise this is a simple solution, but what would another method be?

 

[Carrotsticks]

I think the 'or otherwise' was aimed at determining the direction of the inequality, not to actually solve the entire problem.

And the graphical method is the easiest (and safest more often than not for those not 100% confident with inequalities) method to use to determine it.

 

[shongaponga]

Okay, fair enough. Thanks Carrot