Summation of series

Need to know

What you need to know

• To sum a polynomial like $\sum (ar^3 + br^2 + cr + d)$, split it up: $a\sum r^3 + b\sum r^2 + c\sum r + \sum d$.
• Remember that $\sum_{r=1}^{n} d = nd$.
• If a sum does not start at $r=1$, for example $\sum_{r=a}^{n} u_r$, calculate it as $\sum_{r=1}^{n} u_r - \sum_{r=1}^{a-1} u_r$.

Explanation

The Domino Demolition

1. Express the general term of the series, $u_r$, in the form $f(r) - f(r+1)$. This is often done using partial fractions.
2. Write out the first few terms of the sum (e.g., for r=1, r=2, r=3) and the last few terms (e.g., for r=n-1, r=n).
3. Observe the 'telescoping' cancellation. The $+f(r+1)$ part of one term cancels with the $-f(r)$ part of the next.
4. Add together the remaining terms that did not cancel. This will give you the formula for the sum to n terms, $S_n$.