I managed the first parts, its the last part im having trouble with. (let x be theta)

This part was fine and the sum of the convergence series wasAn infinite series is given by z - z^2 + z^3 - z^4

i.) Assuming the series converges, find an expression for the sum.

ii.) Given that z = 1/4(cos x + isin x) explain why the series converges for all values of x.

So i assume we use this for part b.)

b.) By using de moivres theorm or otherwise, prove that the sum of the infinite series

Converges into:

Well im not sure how to do this by de moivres theorm...but by series:

Sum of infinite series:

So for a being the first in the series, this must be the imaginary part of

And r, the common ration must be imaginary part of

So the convergence is imaginary part of:

Which can be re-arranged to get:

Well the numerator is the same as required in the question however im not sure how to get the denominator...or in fact if i have even done this right.

Any help would be greatley appriciated

Thankyou