Change of variables for summation fails Suppose I have the sum $ \displaystyle S_1 = \sum_{i=1}^5 \cfrac{1}{(i+1)^2} $
I can make a change of variables by letting $ j=i+1 $. When $ i=1 \implies j=2 $, and when $ i=5 \implies j=6 $
$ \displaystyle \therefore S_1 = \sum_{j=2}^6 \cfrac{1}{j^2} $.
But if I were to make a change of variables with $ j=(i+1)^2 $ and use the same logic as above, the resulting sum would be: $ \displaystyle \sum_{j=4}^{36}\cfrac{1}{j} $. Which is not equal to the original sum $ S_1 $.
In theory, why does this change of variable fail?

$\displaystyle \sum_{j=4}^{36}\cfrac{1}{j}=\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+....+\frac{1}{36}$ whereas $\displaystyle \sum_{j=2}^{6}\cfrac{1}{j^2}=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}=\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}$. So their sums are not matching. Instead you can write $$\displaystyle \sum_{j=2}^{6}\cfrac{1}{j^2}=\displaystyle\sum_{j\in\{4,9,16,25,36\}}\cfrac{1}{j}$$. $\displaystyle\sum_{j\in\{4,9,16,25,36\}}\cfrac{1}{j}$ means that it is the sum of all values $\frac{1}{j}$ where $j$ belongs to the set {$4,9,16,25,36$}. This implies that $\displaystyle\sum_{j\in\{4,9,16,25,36\}}\cfrac{1}{j}$ becomes $\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}$ which is the required value.
