when you are checking to see if a sum of say $k^2$ from $k=1$ to to $k=n$ is equal to a sum of $(k+1)^2$ from $k=0$ to $n−1$ can someone explain what is going on here. THanks
(looking for a fairly simple way to work the problem without writing out the sums which may help me understand what is going on, )
On
Think about what summation notation means. The sum $$\sum_{k=1}^n k^2$$ is evaluated by adding a term for each $k = 1, \ldots, n$, and the term for the given value of $k$ is $k^2$. So, the sum becomes $$\sum_{k=1}^n k^2 = 1^2 + 2^2 + 3^2 + \ldots + n^2.$$ Similarly, $$\sum_{k=0}^{n-1} (k + 1)^2 = (0 + 1)^2 + (1 + 1)^2 + \ldots + (n - 1 + 1)^2 = \sum_{k=1}^n k^2.$$
$$S=\sum_{k=1}^nf(k)=\sum_{k=1}^nf(k-i+i)$$
put $k-i=K$ then
$$S=\sum_{K=1-i}^{n-i}f(K+i)$$
you can take $i=1$.