Non sum form of an infinite Sum of polar number

Question

I'm looking for a non sum form of the equation

$\sum_{n=1}^\infty(ba^{n+1}\angle n\theta^{\circ})$

I've been working on this problem for 8 hours and have been unable to make any progress.

So my question is: how can we rewrite this equation to not include an infinite sum?

And also if you would be so kind as tell me the names of the algebraic steps used, so i can look further into them.

Answer 1

You are presumably asking for the sum of the complex numbers $ba^{n+1}e^{in\theta}=ba(ae^{i\theta})^n$.

If true, provided that $|a|<1$ to ensure convergence, by summation of the geometric series,

$$S=\frac{ba^2e^{i\theta}}{1-ae^{i\theta}}=\frac{ba^2(e^{i\theta}-a)}{1-a\cos\theta+a^2}.$$