Sequences and series

What sequences and series covers

Patterns with rules attached, and the two series proofs the specification requires you to be able to reproduce. The binomial theorem expands brackets to any power without multiplying them out, sigma notation keeps the bookkeeping tidy, and arithmetic and geometric series add up the predictable. The general binomial expansion later combines with partial fractions from Algebra.

The main ideas

• Binomial coefficients from Pascal's triangle or from the nCr formula, and (a + bx)ⁿ expanded for positive integer n.
• Pulling one coefficient out of an expansion without writing the rest, and truncating to approximate a power of a number close to 1.
• The general binomial expansion for negative and fractional n, with the range of validity written as an inequality in x.
• Sequences from an nth-term formula or from a recurrence, classified as increasing, decreasing or periodic, and sums written in sigma notation.
• Arithmetic sequences and series, with the forwards-plus-backwards proof of the sum formula.
• Geometric sequences and series, with the subtraction proof, the sum to infinity where the modulus of r is below 1, and logarithms for the how-many-terms questions.

The results it turns on

(a + b)ⁿ = Σ nCr a^n−rb^r

the binomial theorem, for positive integer n

(1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + …, for |x| < 1

the general binomial expansion, for negative or fractional n

u_n = a + (n − 1)d, and S_n = (n/2)(2a + (n − 1)d)

an arithmetic sequence and its sum

u_n = arⁿ⁻¹, and S_n = a(1 − rⁿ)/(1 − r)

a geometric sequence and its sum

S_∞ = a/(1 − r), for |r| < 1

the sum to infinity of a convergent geometric series

Where it usually goes wrong

• For (a + bx)ⁿ the aⁿ has to come out of the bracket first, and the validity condition becomes |bx/a| < 1. Stating that inequality is a mark separate from the expansion itself.
• Reaching the nth term of an arithmetic sequence takes n - 1 steps, so the bracket is written before any numbers go into it.
• Recovering the ratio from two non-adjacent terms means dividing rather than subtracting. If the sixth term divided by the third is 8 then r³ = 8.
• Sigma limits that start above 1 mean subtracting two sums, so terms from 5 to 20 give S_20 minus S_4.

Where to start

The two binomial lessons work as a pair, though the general expansion is better left until partial fractions is secure. Sequences and sigma notation next, since arithmetic and geometric series are both written in it. Geometric series last: it carries the sum to infinity and the logarithm questions, and it is the longest of the five.