Summation of series — common mistakes

In method of differences questions, it is essential to show your working for the cancellation. Write out at least the first two and the last term explicitly. A common mistake is incorrect cancellation, especially if the difference is of the form $f(r) - f(r+2)$. Writing the terms out clearly prevents this.

Standard formulae are used when the general term is a polynomial in $r$ (e.g., $r^3 + 2r$). The method of differences is used for terms that can be written as a difference, like $f(r) - f(r+1)$, which is common for rational functions (fractions with polynomials).

You use the principle of inclusion-exclusion. Calculate the sum from 1 to the upper limit and subtract the sum from 1 to the term before the lower limit. So, $\sum_{r=5}^{10} u_r = \sum_{r=1}^{10} u_r - \sum_{r=1}^{4} u_r$. This applies to both standard formulae and the method of differences.

The question will usually guide you. Often, you'll first be asked to 'Show that' the term can be expressed using partial fractions, which naturally creates the difference form. If you have to find it yourself, look for patterns. For example, $r(r+1)!$ can be written as $(r+1-1)(r+1)! = (r+2)! - (r+1)!$.

While a calculator can compute a numerical sum for a given $n$, exam questions almost always require you to find an algebraic expression for $S_n$ in terms of $n$. You must show the substitution of the standard formulae and the subsequent algebraic simplification and factorisation to earn the marks.