A-level Mathematics/AQA/MPC2

Sequences and series

Notation

$u_{n}\,\!$ — the general term of a sequence; the n^th term

$a\,\!$ — the first term of a sequence

$l\,\!$ — the last term of a sequence

$d\,\!$ — the common difference of an arithmetic progression

$r\,\!$ — the common ratio of a geometric progression

$S_{n}\,\!$ — the sum to n terms: $S_{n}=u_{1}+u_{2}+u_{3}+\ldots +u_{n}\,\!$

$\sum \,\!$ — the sum of

$\infty \,\!$ — infinity (which is a concept, not a number)

$n\rightarrow \infty \,\!$ — n tends towards infinity (n gets bigger and bigger)

$|x|\,\!$ — the modulus of x (the value of x, ignoring any minus signs)

Convergent, divergent and periodic sequences

Convergent sequences

A sequence is convergent if its n^th term gets closer to a finite number, L, as n approaches infinity. L is called the limit of the sequence:

${\mbox{As }}n\to \infty {\mbox{, }}u_{n}\to L\,\!$

Another way of denoting the same thing is:

$\lim _{n\to \infty }u_{n}=L\,\!$

Definition of the limit of a convergent sequence

Generally, the limit $L\,\!$ of a sequence defined by $u_{n+1}=f(u_{n})\,\!$ is given by $L=f(L)\,\!$

Divergent sequences

Sequences that do not tend to a limit as $n$ increases are described as divergent. eg: 1, 2, 4, 8, 16, ...

Periodic sequences

Sequences that move through a regular cycle (oscillate) are described as periodic.

Series

A series is the sum of the terms of a sequence. Those series with a countable number of terms are called finite series and those with an infinite number of terms are called infinite series.

Arithmetic progressions

An arithmetic progression, or AP, is a sequence in which the difference between any two consecutive terms is a constant called the common difference. To get from one term to the next, you simply add the common difference:

$u_{n+1}=u_{n}+d\,\!$

Expression for the n^th term of an AP

$u_{n}=a+(n-1)d\,\!$

Formulae for the sum of the first n terms of an AP

The sum of an arithmetic progression is called an arithmetic series.

$S_{n}={\frac {n}{2}}\left\lbrack 2a+(n-1)d\right\rbrack \,\!$

$S_{n}={\frac {n}{2}}(a+l)\,\!$

Formulae for the sum of the first n natural numbers

The natural numbers are the positive integers, i.e. 1, 2, 3…

$S_{n}={\frac {n}{2}}(n+1)\,\!$

Geometric progressions

An geometric progression, or GP, is a sequence in which the ratio between any two consecutive terms is a constant called the common ratio. To get from one term to the next, you simply multiply by the common ratio:

$u_{n+1}=ru_{n}\,\!$

Expression for the nth term of an GP

$u_{n}=ar^{n-1}\,\!$

Formula for the sum of the first n terms of a GP

$S_{n}=a\left({\frac {1-r^{n}}{1-r}}\right)\,\!$

$S_{n}=a\left({\frac {r^{n}-1}{r-1}}\right)\,\!$

Formula for the sum to infinity of a GP

$S_{\infty }=\sum _{n=1}^{\infty }ar^{n-1}={\frac {a}{1-r}}\qquad {\mbox{where }}-1<r<1\,\!$

Binomial theorem

The binomial theorem is a formula that provides a quick and effective method for expanding powers of sums, which have the general form $(a+b)^{n}$.

Binomial coefficients

The general expression for the coefficient of the $(r+1)^{th}$ term in the expansion of $(1+x)^{n}$ is:

${}^{n}\!C_{r}={\binom {n}{r}}={\frac {n!}{r!(n-r)!}}$

where $n!=1\times 2\times 3\times \ldots \times n$

$n!$ is called n factorial. By definition, $0!=1$.

Binomial expansion of (1+x)ⁿ

$(1+x)^{n}=1+{\binom {n}{1}}x+{\binom {n}{2}}x^{2}+{\binom {n}{3}}x^{3}+\ldots +x^{n}$

$(1+x)^{n}=1+nx+{\frac {n(n-1)}{2!}}+{\frac {n(n-1)(n-2)}{3!}}+\ldots +x^{n}$

$(1+x)^{n}=\sum _{r=0}^{n}{\binom {n}{r}}x^{r}$

Trigonometry

Arc length

$l=r\theta \,\!$

Sector area

$A={\tfrac {1}{2}}r^{2}\theta$

Trigonometric identities

$\tan {\theta }\equiv {\frac {\sin {\theta }}{\cos {\theta }}}$

$\sin ^{2}{\theta }+\cos ^{2}{\theta }\equiv 1\,\!$

Indices and logarithms

Laws of indices

$x^{m}\times x^{n}=x^{m+n}\,\!$

$x^{m}\div x^{n}=x^{m-n}\,\!$

$\left(x^{m}\right)^{n}=x^{mn}\,\!$

$x^{0}=1\,\!$ (for x ≠ 0)

$x^{-m}={\frac {1}{x^{m}}}\,\!$

$x^{\frac {1}{n}}={\sqrt[{n}]{x}}\,\!$

$x^{\frac {m}{n}}={\sqrt[{n}]{x^{m}}}\,\!$

Logarithms

$10^{2}=100\Leftrightarrow \log _{10}{100}=2$

$10^{3}=1000\Leftrightarrow \log _{10}{1000}=3$

$2^{5}=32\Leftrightarrow \log _{2}{32}=5$

$\log _{a}{b}=c\Leftrightarrow a^{c}=b$

Laws of logarithms

The sum of the logs is the log of the product.

$\log {x}+\log {y}=\log {xy}\,\!$

The difference of the logs is the log of the quotient.

$\log {x}-\log {y}=\log {\left({\frac {x}{y}}\right)}$

The index comes out of the log of the power.

$k\log {x}=\log {\left(x^{k}\right)}$

Differentiation

Differentiating the sum or difference of two functions

$y=f(x)\pm g(x)\quad \therefore \quad {\frac {dy}{dx}}=f'(x)\pm g'(x)$

Integration

Integrating axⁿ

$\int ax^{n}\,dx={\frac {ax^{n+1}}{n+1}}+c\qquad {\mbox{ for }}n\neq -1\,\!$

Area under a curve

The area under the curve $y=f(x)$ between the limits $x=a$ and $x=b$ is given by

$A=\int _{a}^{b}y\,dx$