Pure: Sequences, Series and Binomial Depth

Method

1. For (1+x)ⁿ with n a negative or non-integer rational number, use the binomial series (1+x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + ..., an infinite series valid only for |x| < 1 - always state this range as part of the answer.For (1+x)^n with n a negative or non-integer rational number, use the binomial series (1+x)^n = 1 + nx + [n(n-1)/2!]x^2 + [n(n-1)(n-2)/3!]x^3 + ..., an infinite series valid only for |x| < 1 - always state this range as part of the answer.
2. For (a+bx)ⁿ where a is not 1, write it as aⁿ(1+bax)ⁿ first, expand the bracket using the standard series in bax, then multiply every term by aⁿ; the validity range becomes |x| < |a/b|.For (a+bx)^n where a is not 1, write it as a^n(1+(b/a)x)^n first, expand the bracket using the standard series in (b/a)x, then multiply every term by a^n; the validity range becomes |x| < |a/b|.
3. If the expression to expand is a rational function (for example from a partial-fractions decomposition), expand each partial fraction separately as its own binomial series, then collect like powers of x; the overall validity range is the more restrictive (smaller) of the individual ranges.
4. For σ-notation sums, split a sum of several terms into separate standard sums (such as a sum of r² minus a multiple of a sum of r plus a constant sum), and substitute the standard formulae [n(n+1)/2 for sum of r, n(n+1)(2n+1)/6 for sum of r²] before simplifying algebraically.For sigma-notation sums, split a sum of several terms into separate standard sums (such as a sum of r^2 minus a multiple of a sum of r plus a constant sum), and substitute the standard formulae [n(n+1)/2 for sum of r, n(n+1)(2n+1)/6 for sum of r^2] before simplifying algebraically.
5. To identify what kind of sequence a recurrence relation generates, compute the first few terms by hand: a constant difference means arithmetic (use u_n = a+(n-1)d), a constant ratio means geometric (use u_n = ar^n-1, checking |r| < 1 before assuming a sum to infinity exists), and terms that repeat means periodic (identify the period, then use it to jump straight to any required term).To identify what kind of sequence a recurrence relation generates, compute the first few terms by hand: a constant difference means arithmetic (use u_n = a+(n-1)d), a constant ratio means geometric (use u_n = ar^(n-1), checking |r| < 1 before assuming a sum to infinity exists), and terms that repeat means periodic (identify the period, then use it to jump straight to any required term).
6. For a geometric series sum to infinity, first justify convergence by checking |r| < 1 (if the question does not already state it), then apply S_infinity = a/(1-r).For a geometric series sum to infinity, first justify convergence by checking |r| < 1 (if the question does not already state it), then apply S_infinity = a/(1-r).
7. When a 'show that' question gives a recurrence and a claim about a term or a sum, work directly from the recurrence (substituting the given u_n and simplifying algebraically) rather than guessing and checking, since this generalises to any term the question might later ask about.When a 'show that' question gives a recurrence and a claim about a term or a sum, work directly from the recurrence (substituting the given u_n and simplifying algebraically) rather than guessing and checking, since this generalises to any term the question might later ask about.

Practice questions

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