Sigma notation, sequences, series, finance

Sigma notation
- properties
Arithmetic and geometric sequences and series

Sigma notation

Summation usually specify lower and upper limits, using indices such as $i$ and $j$ .

\sum_{i = 2}^5{i}=2+3+4+5=14

\sum_{j = 3}^\infty 3^{-j}=3^{-3} + 3^{-4} + \dots

Summation may also use a variable to denote all possible values.

\sum_{k} u_k = \text{sum over all possible values of } k

There are implicit brackets around the first term after the $\sum$

\sum_{m} \frac{3m^2 + 5}{m -1} + 4 = \left(\sum_{m} \frac{3m^2 + 5}{m -1}\right) + 4

properties

Summation is just addition, so all properties of addition apply for summation.

$u_n$ and $v_n$ are arbitrary expressions of $n$ . $k$ and $N$ are constants that do not depend on $n$ .

\begin{align*} &\sum_{n = 0}^5 u_n = \sum_{n = 0}^3 u_n + \sum_{n = 4}^5 u_n \\ &\sum_{n = 3}^N (u_n \pm v_n) = \sum_{n = 3}^N u_n \pm \sum_{n = 3}^N v_n \\ &\sum_{n = 2}^\infty ku_n = k\sum_{n = 2}^\infty u_n \end {align*}

Example: Expand

\frac1k\sum_{i = 1}^k (x_i - \mu)^2

with

\mu = \frac1k\sum_{i = 1}^k x_i

\begin{align*} \text{sum} &= \frac1k\sum_{i = 1}^k (x_i - \mu)^2 \\ &= \frac1k\sum_{i = 1}^k \left({x_i^2 -2x_i\mu + \mu^2}\right) \\ &= \frac1k\sum_{i = 1}^k x_i^2 -\frac{2\mu}{k}\sum_{i = 1}^k x_i + \frac{\mu^2}{k}\sum_{i = 1}^k 1 \\ &= \frac1k\sum_{i = 1}^k x_i^2 -\frac{2\mu}{k}(\mu k) + \frac{\mu^2}{k}k \\ &= \frac1k\sum_{i = 1}^k x_i^2 -2\mu^2 + \mu^2 \\ &= \frac1k\sum_{i = 1}^k x_i^2 -\mu^2 \qed \end {align*}

The only new property is a change of variable. For example, making a substitution of $m = n - 1$ . Note both limits change, and $n - 1$ is replaced by $m$ , and that $7$ is still $7$ .

\sum_{n=1}^97^{n-1} = \sum_{m=0}^87^m

Arithmetic and geometric sequences and series

definitions

term - an expression involving some index (order), typically $n$ . $u_1$ means first term; $u_n$ means $n$ th term

sequence - a list of terms in order

series - the summation of some terms

infinite series - the summation of an infinite number of terms

arithmetic successive terms differ by a constant difference or common difference, eg $3, 5, 7, 9$

geometric successive terms differ by a constant(common) factor(ratio), eg $2, -6, 18, -54 \dots$

converging series - an infinite series with some finite, deterministic value

diverging series - an infinite series that goes to positive or negative infinity, or it oscillates with constant amplitude.

formulas

	arithmetic	geometric
explicit definition	${u_n = u_1+d(n-1)}$	$u_n = u_1r^{n-1}$
recursive definition	${u_n = u_{n-1}+d}$	$u_n = u_{n-1}r$
finite series	$\displaystyle {S_n = \frac{n}2(u_1 + u_n)} \\ {S_n = \frac{n}2(2u_1 + (n-1)d)}$	$\displaystyle {S_n = \frac{u_1 - u_{n+1}}{1-r}} \\ {S_n = \frac{u_1(1 - r^n)}{1 - r}}$
infinite series		$\displaystyle S_\infty = \frac{u_1}{1 - r},\;\; \lvert r\rvert < 1$

estimation

Sometimes, a common difference for arithmetic sequences and series need to be estimated. One way to do so is

d_\text{est.} = \frac{u_n - u_1}{n - 1}

which is a rearrangement of the explicit definition of an arithmetic sequence.

tips

From $0$ to $9$ , there are $9 - 0 + 1 = 10$ numbers.

Example: Find the sum of all multiples of $7$ from $20$ to $200$ .

$\begin{align*} 21 &= 7 \times 3 \\ 196 &= 7 \times 28 \\ \end{align*}$
There are $n = 28 - 3 + 1 = 26$ terms

It’s an arithmetic series, because each term is $7$ more than the previous, and we need the sum.
$S_{26} = \frac{26}{2}(21 + 196) = 2821 \qed$
Each term can be expressed with any other term, not only $u_1$ .

Example: Three consecutive terms in a geometric sequence multiply to 64. Find the middle term.

Let the middle term be $x$ and common ratio be $r$ . The three terms are $\frac xr, x, xr$ .

Their product is $64 = \left(\frac xr \right)(x)(xr) = x^3$ . So $x = 4.\qed$

Example: In a sequence of real numbers, the sixth term is 16 less than the fourth term. The seventh term is half of the third term. Find the tenth term if

(a) it is an arithmetic sequence.

(b) it is a geometric sequence.

(a)

Let the third term be $x$ , common difference be $d$ .

“sixth term is 16 less than the fourth term”
$\begin{align*} (x+d) + 2d &= (x+d) - 16 \\ d &= -8 \end{align*}$
“seventh term is half of the third term”
$\begin{align*} x + 4d &= \frac x2 \\ \frac x2 &= -4d \\ x &= -8d \\ &= 64 \\ \end{align*}$ $u_{10} = x + 7d = 64 - 7(8) = 8 \qed$
Note, the number of common ratio or difference between $m$ th and $n$ th terms is just $n - m$ , without the $+1$ , because it’s just differences.

(b)

Let the fourth term be $x$ , common ratio be $r$ .

“seventh term is half of the third term”
$\begin{align*}xr^3 &= \frac {x}{2r} \\ r^4 &= \frac 12 \\ r^2 &= \frac{\sqrt 2}{2} \\ \end{align*}$
“sixth term is 16 less than the fourth term”
$\begin{align*}(x) r^2 &= (x) - 16 \\ x (r^2 - 1) &= -16 \\ x &= \frac{-16 (2)}{\sqrt{2} - 2} \\ x &= \frac{-32 (- \sqrt 2 - 2)} 2 \\ x &= 16\sqrt 2 + 32 \end{align*}$ $\begin{align*} u_{10} = xr^6 &= (16\sqrt 2 + 32)\frac{\sqrt 2} 4\\ u_{10} &= 8 + 8\sqrt2 \qed \end{align*}$

compound interest

FV = PV \left(1+\frac{r}{100k}\right)^{kn}

$FV$ : future value, valuation after $n$ years after interest accumulates

$PV$ : present value, valuation before interest accumulates

$r$ : The annual (nominal) interest rate is $r\%$

$k$ : compounding periods per year

$k$	compounding period
$1$	annually (yearly)
$2$	every $6$ months (half-yearly)
$4$	every $3$ months (quarterly)
$12$	every month (monthly)

Example: To save $20,000 over $4$ years with $5\%$ annual interest compounded monthly, what should be the lump sum amount to be invested today?

\begin{align*} FV &= 20000 \\ n &= 4\\ r &= 5\\ k &= 12\\ PV &= \frac{FV}{\left(1+\frac{r}{100k}\right)^{kn}} \\ PV &= \frac{20000}{\left(1+\frac{5}{100(12)}\right)^{12(4)}} \\ PV &= \$16381.42 \qed \end{align*}

Financial question often says how many significant figures to keep. If it does not say, keep two decimal places.

Appreciation means it becomes more valuable. Depreciation means it loses value. For depreciation, $r < 0$ .

See usages on TI-84 Plus. Beware of the differences between payment period and compounding period, and the tendency to introduce negative numbers.

real value with inflation

Inflation is when money loses value (which may or may not be exacerbated in recent times by excessive money-printing).

The real interest rate, which reflects the increase in real value of money, is defined as

\text{real interest rate} = \left(\frac{100\% + \text{nominal interest rate}}{100\% + \text{inflation rate}} - 1\right) \cdot 100\%

In contrast, the nominal interest rate is the listed interest rate on paper, without considering inflation.

Tip: IB also accepts

{\text{real interest rate} = \text{nominal interest rate} - \text{inflation rate}}

Example: Under $3\%$ annual inflation, a fund yields an $8\%$ annual interest rate compounded yearly. A person invests $\$10,000$ on Jan 1, 2024 into this fund. To the nearest dollar, find the value of this investment on Jan 1, 2030

i. in year 2030 dollar amounts;

ii. in year 2024 dollar amounts.

i. This is compounded interest with $PV$ = $10000$ , $r = 0.08$ , $k = 1$ and $n = 6$ . Unless specified otherwise, all interest rates are nominal.

\begin{align*} FV &= 10000\left(1+\frac{0.08}{1}\right)^{1(6)} \\ &\approx 15869 \quad (2030 \text{ dollar})\qed\end{align*}

ii. The real interest rate is

\left(\frac{1.08}{1.03} - 1\right)\cdot 100\% \approx 4.85437\%

\begin{align*} FV &= 10000\left(1+\frac{0.0485437\dots}{1}\right)^{1(6)} \\ &\approx 13290 \quad (2024 \text{ dollar})\qed\end{align*}

This is numerically equal to

10000\left(\frac{1.08}{1.03}\right)^6 = \frac{15869}{(1.03)^6}

The real interest rate is an estimate of the true annual interest rate. It can then be used as a regular annual interest rate in the compound interest formula.