Sequences and Series
Core concepts
You must handle arithmetic and geometric progressions, sums to n terms, the conditions for convergence of a geometric series, and the binomial expansion for positive integer and for rational indices. Know the difference between a sequence (ordered list) and a series (sum), and be confident with sigma notation and simple method-of-differences results for sums of powers.
Traps
- The sum to infinity exists only when |r| < 1; stating S∞ = a/(1 − r) without that condition loses marks.
- In a geometric context, an unknown ratio may have two valid values (e.g. from a quadratic); check which satisfy the question.
- For the binomial expansion with a fractional or negative index, the expansion is valid only for |x| < 1 after the correct factorisation. Always factor out the constant so the second term is of the form (1 + y) with |y| < 1.
- Confusing the nth term with the sum of n terms is a common slip.
Worked example
Expand (1 + 2x)^(−1) up to the x² term. Using the general result, (1 + y)^n = 1 + ny + n(n−1)y²/2! + … with n = −1, y = 2x: 1 + (−1)(2x) + (−1)(−2)(2x)²/2 = 1 − 2x + 4x². Valid for |2x| < 1, i.e. |x| < ½. Rule of thumb: write the validity condition as soon as you finish expanding.