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.

$\displaystyle

\frac{z}{1 + z}

$

So i assume we use this for part b.)

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

\frac{sin x }{4} - \frac{sin 2x}{4^2} + \frac{sin 3x}{4^3}...(-1)\frac{sin nx}{4^n}

$

Converges into:

$\displaystyle

\frac{4sinx}{17 + 8cos x}

$

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

Sum of infinite series:

$\displaystyle \frac{a}{1 - r}$

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

$\displaystyle \frac{e^ix}{4}$

And r, the common ration must be imaginary part of

$\displaystyle \frac{- e^ix}{4}$

So the convergence is imaginary part of:

$\displaystyle

\frac{\frac{e^ix}{4}}{1 + \frac{e^ix}{4}}

$

Which can be re-arranged to get:

$\displaystyle

\frac{4sin x}{16 + 4sin x}

$

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