Financial Mathematics Text

Sunday, August 11, 2013

Financial Mathematics: Sequences and Series

In mathematics, a sequence is an ordered list of numbers. A series is the sum of the terms of a sequence. Series are also a kind of sequence. Series are an essential tool in dealing with certain kinds of financial instruments.

Here's a quick example:

Let $a$ be the sequence $(1, 2, 3)$. Then $a_0 = 1$, $a_1=2$ and $a_2=3$. The subscript denotes which place in the sequence we're at (with zero being the first).

We can construct a series from this sequence by adding the terms. We'll denote the first term in the series as $s_0$ and it will simply be the sum of terms up to the first term:
$$s_0 = a_0 = 1$$
We can then look at the second term, $s_1$, as the sum of the first two terms in the sequence:
$$s_1 = a_0 + a_1 = 1 +2 =3$$
Lastly, we can look at the third (and final term) of this sequence:
$$s_2 = a_0 + a+1 + a_2 = 1 + 2 + 3 =6$$
So the series we just defined is $s=(1,3,6)$.

Some sequences and series are finite (such as the example above) while others are infinite.

One way we can notate a series (or any summation) is by using a big Greek sigma which represents summation:
$$\sum_{i=0 }^{n} a_i =  a_0 + a_1 + . . . + a_n$$
This simply says to add up all of the $a_i$'s from 0 to $n$.  In the case of infinite series we get this:
$$\sum_{i=0 }^{\infty} a_i = a_0 + a_1 + a_2 + . . .$$
With respect to the infinite series we can ask the following question: Does the series converge or does it diverge?

A series that converges approaches a specific finite value. For example, the Harmonic Sequence converges to 0:

Harmonic Sequence:
$$\left(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5},. . .\right)$$
The Harmonic Series, however, diverges. If you sum up all of the terms, it goes to infinity (and beyond!) We can represent the harmonic series as a summation:
$$\sum_{i=1}^{\infty} \frac{1}{i}= \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + . . .$$
Not all sequences that diverge go to infinity. Some are said to oscillate like this one:

$$(1, -1, 1, -1, 1, -1, 1, -1,. . .)$$
It doesn't converge to a finite value but it doesn't go anywhere specific either. It just bounces back and forth between 1 and -1. (Notice, that the series diverges as well, but it bounces back and forth between 1 and 0.)

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