Partial Fractions as Power Series

Question

I have the partial fraction sum

$$f(i\omega)= a_0 + \frac{a_1}{\lambda_1+i\omega} + \frac{a_2}{\lambda_2+i\omega}$$

Which I want to represent as a power series in $ x = i\omega $

I thought that the solution was

$$f(x) = a_0 + a_1\sum_{k=0}^{\infty}-\lambda_1^kx^k + a_2\sum_{k=0}^{\infty}-\lambda_2^kx^k$$

However, clearly I am doing something wrong because the solution is not working. Can anyone point out my mistake? I am not familiar with power series expansions so apologies if it is a simple question.

Thanks so much.

Answer 1

$$\frac{1}{\lambda_1 + x} = \frac{1}{\lambda_1}\cdot\frac{1}{1+\frac{x}{\lambda_1}} = \frac{1}{\lambda_1}\sum_{i=0}^\infty (-1)^i\left(\frac{x}{\lambda_1}\right)^i.$$

Answer 2

$$\dfrac{1}{\lambda + x} = \dfrac{1}{\lambda (1 + x/\lambda)} = \dfrac{1}{\lambda} - \dfrac{x}{\lambda^2} + \dfrac{x^2}{\lambda^3} - \ldots $$