Lets say we have the convergent sum $$ \ \sum_{n=1}^{\infty} a_n = X \ $$ $$ \ X= \sqrt[]{2}-1 $$
And we know that for every value of n: $$ a_n>0 $$
And you want to find out what Y is in terms of X. $$ \ \sum_{n=1}^{\infty} (a_n)^2=Y\ $$
Square root on both sides gives: $$ \sqrt[]{\sum_{n=1}^{\infty} (a_n)^2}=\sqrt[]{Y}\ $$
We also know that Y is convergent.
Now the left side looks like the equation for distance in an n-dimensional space. So my idea was that if I knew how to rotate a point in n dimensions such that the distance from the center is preserved, I could find a point on that sphere for which the sum of its coordinates converges and its easy to calculate. That way I could find out what the value of Y is. My problem is that i have no idea how to rotate a point in N dimensions. Is the idea even worth trying?
You can't find $Y$ as a function of $X$. Suppose $$ a_1 = \sqrt{2} -1 $$ and all the other values of $a_n$ are $0$. Suppose $$ b_1 = \sqrt{2}\ \text{ and }\ b_2 = -1 $$ and all the other values of $b_n$ are $0$.
Then these two sequences have the same $X$ but different $Y$'s.