General summation formula for...

Question

I'm looking for a general summation formula for $\left((1-\lambda)\lambda^0+(1-\lambda)\lambda^1\right)R^{(2)} + \left((1-\lambda)\lambda^2+(1-\lambda)\lambda^3\right)R^{(4)} + \left((1-\lambda)\lambda^4+(1-\lambda)\lambda^5\right)R^{(6)} + \ldots$

Answer 1

You mean something like this?

$$\sum _{i=0}^{\infty} (1-\lambda)({\lambda}^{2i}+{\lambda}^{2i+1})R^{(2i+2)}$$

As pointed out in the comments, you could factor the $(1-\lambda)$ term out of the sum as:

$$(1-\lambda)\sum _{i=0}^{\infty} ({\lambda}^{2i}+{\lambda}^{2i+1})R^{(2i+2)}$$

Answer 2

This sum can be written

$$(1-\lambda^2)\sum_{k=0}^\infty \lambda^{2k}R^{(2k+2)}.$$

If $R^{(n)}$ is to be understood as $R^n$, this geometric series converges to

$$\frac{(1-\lambda^2)R^2}{1-\lambda^2R^2}$$ when $|\lambda R|<1$.