H2 Maths Notes (JC 1-2): 2.1) Sequences and Series

Before you revise\ Keep a separate page for identities: arithmetic/geometric sums, binomial expansions, and standard limits. Most errors come from mixing up first term and common ratio definitions.

Core Definitions

• A sequence is an ordered list of terms \( {u_n} \). The nth partial sum is \( S_n = u_1 + u_2 + \dots + u_n \).
• Arithmetic progression (AP): constant difference \( d \) with \( u_n = a + (n - 1)d \) and \( S_n = \tfrac{n}{2}\bigl(2a + (n - 1)d\bigr) \).
• Geometric progression (GP): constant ratio \( r \) with \( u_n = ar^{n - 1} \) and \( S_n = a \frac{1 - r^n}{1 - r} \) for \( r \neq 1 \).
• Infinite GP converges if \( \lvert r \rvert < 1 \) with sum \( S_\infty = \frac{a}{1 - r} \).

Convergence and Limits

• Use limits to test behaviour: \( \lim\limits_{n \to \infty} u_n \) exists if terms settle to a finite value.
• For recursive sequences \( u_{n+1} = f(u_n) \), solve the fixed point \( L = f(L) \) and show \( {u_n} \) is monotonic and bounded.
• Divergence occurs when terms grow without bound or oscillate without settling.

Example -- Recurrence limit

Given \( u_1 = 2 \) and \( u_{n+1} = \frac{1}{2}\bigl(u_n + \frac{5}{u_n}\bigr) \), show convergence.

1. Suppose \( u_n \to L \); then \( L = \frac{1}{2}\bigl(L + \frac{5}{L}\bigr) \) leading to \( L^2 = 5 \) so \( L = \sqrt{5} \) (take positive root since terms stay positive).
2. Show \( u_n \) is increasing and bounded above by \( \sqrt{5} \).

Sigma Notation and Manipulation

• Express sums as \( \sum\limits_{k=1}^n u_k \). Familiar identities: \[ \sum_{k=1}^n k = \frac{n(n + 1)}{2}, \qquad \sum_{k=1}^n k^2 = \frac{n(n + 1)(2n + 1)}{6}, \qquad \sum_{k=1}^n r^k = r \frac{1 - r^n}{1 - r}. \]
• Split linear combinations: \( \sum (a_k + b_k) = \sum a_k + \sum b_k \).
• To shift indices, substitute \( k = j + c \) and adjust limits.

Example -- Telescoping sum

Evaluate \( \sum\limits_{k=1}^n \frac{1}{k(k + 1)} \).

1. Decompose: \( \frac{1}{k(k + 1)} = \frac{1}{k} - \frac{1}{k + 1} \).
2. Terms telescope to \( 1 - \frac{1}{n + 1} = \frac{n}{n + 1} \).

Arithmetic and Geometric Applications

• Mixture problems: combine AP and GP components for salaries, payments, or depreciation.
• Sum to infinity: verify \( \lvert r \rvert < 1 \) before using \( S_\infty \).
• Insert means: For inserting \( k \) arithmetic means between \( a \) and \( b \), use AP formula; for geometric means, use GP ratio \( r = \bigl(\frac{b}{a}\bigr)^{1/(k+1)} \).

Example -- Investment

A fund receives \( \pu{200} \) monthly with constant growth of \( 1.2\%\). After 36 deposits, the total amount is an increasing annuity: \[ S_{36} = 200 \times \frac{(1.012)^{36} - 1}{0.012} \approx \pu{8.56 \times 10^3}. \]

Binomial Links

• Use binomial coefficients: \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \).
• Expand \( (1 + x)^n \) and match coefficients to sequences.
• Recognise common combinatorial sums like \( \sum \binom{n}{k} = 2^n \).

Calculator Workflow

• GC SEQ mode generates recursive terms quickly; verify monotonicity.
• Use sum( or Σ( functions to compute partial sums while showing the manual formula in working.
• Store ratio/difference in variables to avoid transcription errors.

Exam Watch Points

• Always state whether a GP converges before quoting \( S_\infty \).
• Provide exact forms (fractions, radicals) rather than rounded decimals for general expressions.
• In recurrence proofs, justify both monotonicity and boundedness; do not state convergence without proof.
• Translate word problems into algebra carefully, labelling first term, common difference/ratio, and number of terms.

Quick Revision Checklist

• [ ] Derive AP/GP formulae and apply them to contextual problems.
• [ ] Manipulate sigma notation confidently, including telescoping and index shifts.
• [ ] Analyse convergence of recursive sequences using fixed points.
• [ ] Combine AP and GP reasoning in financial or growth questions.

Next steps: Proceed to Topic 3 for vector fundamentals and three-dimensional geometry.