# Real number in terms of infinite series

This is just continuation of my previous post.

$$A = a_0 + \left(\frac{1}{a_1}\right)^k + \left(\frac{1}{a_2}\right)^k + \left(\frac{1}{a_3}\right)^k +\ldots$$ Where $i\ge1$ and the recurrence relation $a_{i+1}\ge a_i\ge2$.

I just raised for each term by $k$. here $k\ge1$ and $A$ is any given real number. Then the above series will exist. Can we generalize this series of some real number with those initial conditions? If yes, kindly discuss...

edit The first term is $a_0$ but not $a_1$. Of course I edited now.

Edited and extended $$A = a_0 + \left(\frac{1}{a_1}\right)^k + \left(\frac{1}{a_1}\right)^k \left(\frac{1}{a_2}\right)^k + \left(\frac{1}{a_1}\right)^k \left(\frac{1}{a_2}\right)^k\left(\frac{1}{a_3}\right)^k +\ldots$$