Same or different?

Excerpt: Students are asked to compare three sums written in sigma notation and asked if they are the same or different. Generalisation is encouraged.

Problem

What’s the same and what’s different about the following sums?

$\displaystyle\sum_{r=0}^{n} r^2$

$\displaystyle\sum_{r=1}^{n} r^2$

$\displaystyle\sum_{r=0}^{n-1} (r+1)^2$

Things you might have noticed

If we start by thinking about what is written down then we might notice the following:

The limits of each sum are different.
The function being summed is different in (3).
All the functions are quadratic.
Sums (2) and (3) have the same number of terms.

Probably the simplest way to notice more is to start writing out the sums.

$\displaystyle\sum_{r=0}^{n} r^2 = 0^2 + 1^2 + 2^2 + \cdots + n^2$

$\displaystyle\sum_{r=1}^{n} r^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2$

$\displaystyle\sum_{r=0}^{n-1} (r+1)^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2$

We can see that (2) and (3) have exactly the same expansion.

We could think of this as a transformation. The function $r^2$ has been transformed to $(r+1)^2$ , a translation of $1$ unit, and the limits have been similarly transformed.

Also, despite there being an extra term in (1), because it is $0$ , all three sums are actually the same value. So we could write $\displaystyle\sum_{r=0}^{n} r^2 = \displaystyle\sum_{r=1}^{n} r^2 = \displaystyle\sum_{r=0}^{n-1} (r+1)^2.$

How can the limits be changed but the sum remain the same?

How can the function be changed but the sum remain the same?

Can you think of some more examples like this?

What is the difference between these two sums? $\sum_{r=0}^{n} (r+1)^2 - \sum_{r=0}^{n} r^2$

Are there any generalisations we can make about summing series?