Sums of powers and finite series

What you’ll learn

• Use standard sums of integers, squares and cubes.
• Rewrite polynomial summands and shifted ranges.

Understand the idea

Sigma notation tells you both the expression to add and the integer range. A polynomial in the summation variable can be split into standard sums using linearity; constants must be counted once for every term.

When a range begins after $1$, subtract the unwanted initial sum from a sum beginning at $1$. Expand a shifted expression such as $(r+1)^2$ before applying formulas, and check a proposed result with a small value of $n$.

Key formulae

$$\sum_{r=1}^n r=\frac{n(n+1)}2$$

$$\sum_{r=1}^n r^2=\frac{n(n+1)(2n+1)}6$$

$$\sum_{r=1}^n r^3=\frac{n^2(n+1)^2}{4}$$

$$\sum_{r=m}^n f(r)=\sum_{r=1}^n f(r)-\sum_{r=1}^{m-1}f(r)$$

Worked example

Find a formula for $\sum_{r=1}^n r(r+2)$.

1. Expand the summand to $r^2+2r$.
2. Use $\frac{n(n+1)(2n+1)}6+2\frac{n(n+1)}2$.
3. Factor out $n(n+1)/6$ and combine $2n+1+6$.

Answer: $\frac{n(n+1)(2n+7)}6$.