Consider the series
$0+\frac{1}{3}+\frac{1}{2}+\frac{3}{5}+\frac{2}{3}+\frac{5}{7} + ...$
Are there multiple ways to write this in sigma notation? For example:
A) $\sum_{k=0}^{infinity} \frac{k}{k+2}$
B) $\sum_{k=1}^{infinity} \frac{k-1}{k+1}$
Are both equivalent? Or is one better than the other - I came up with the first one whereas as the second one is the given answer. Is there a reason why this is?
They are not both equivalent since $A)$ ends at $\frac{n}{n+2}$ while $B)$ ends at $\frac{n-1}{n+1}$. So $A)$ has one extra term.
Making the change of index of summation $k'=k+1$, the limits at $k=0$ and $k=n$ become $k'=1$ and $k'=n+1$ respectively. Thus we have $$\sum_{k=0}^{n}\frac{k}{k+2}=\sum_{k'=1}^{n+1}\frac{k'-1}{k'-1+2}=\sum_{k'=1}^{n+1}\frac{k'-1}{k'+1}=\sum_{k=1}^{n+1}\frac{k-1}{k+1}$$ since $k'$ is a variable.
This is not unique. Choosing the index of summation $k'= k+j$, for $j\in\mathbb Z$, we have $$\sum_{k=0}^{n}\frac{k}{k+2}=\sum_{k'=j}^{n+j}\frac{k'-j}{k'-j+2}=\sum_{k=j}^{n+j}\frac{k-j}{k-j+2}$$
In the above case we have $j=1$.