Sequences and Series

What are sequences?

In Math, a sequence is a list of numbers arranged in a given order, with a certain pattern.

The following are examples of sequences:

Example 1 – 1, 3, 5, 7, 9, ….

In example 1, the first term is 1, the second term is 3 and so on. The pattern in this sequence involves adding 2 to the previous term to get the next term.

Example 2 – 1, 4, 9, 16, 25, ….

In example 2, the first term is 1, the second term is 4 and so on. The sequence consists of squared numbers.

What are series?

In Math, a series is the sum of numbers in the sequence.

Hence, if we are looking at the sequence 1, 3, 5, 7, 9….., then the series is 1 + 3 + 5+ 7+ 9+…

Finite vs Infinite series

A finite series is one that has a finite number of terms.

Example: A series 1+ 3+ 5+ 7+ 9 + 11 has a finite number of terms (i.e. 6 terms), and hence is a finite series.

An infinite series is one that has an infinite number of terms.

Example: A series 1+ 3+ 5+ 7+ 9+ 11+ … has an infinite number of terms. The +… tells us that the series continues with the given pattern, and there is no end. Hence this is an infinite series.

Ways of defining a sequence

Expressing the term as f(n)

There are many ways of defining a term in a sequence. One way is to express the nth term in the form of f(n), where n refers to which term it is.

For example, for the sequence 1, 3, 5, 7, 9, 11, … we can write the nth term to be 2n-1, where n is an integer greater than or equal to 1.

Expressing the term as a function of the previous term

Another way of expressing a term is to write it as a function of its previous term.

For example: uₙ₊₁ = 2uₙ -1, where u₁ = 2, and n is an integer greater than or equal to 1.

Given this expression, we can find u₂ by using the formula u₂ = 2u₁ -1 = 2(2)-1 = 3. Similarly, subsequent terms can be found if the previous term is known.

Expression for Sum vs Expression for Term

We usually use Sₙ to denote the sum of the first n terms in a series, and uₙ to denote the nth term.

To find uₙ given Sₙ, use the formula uₙ = Sₙ- Sₙ₋₁.

Convergent series

A convergent series is one in which as n → ∞, uₙ→ L, where L is a real number, and we say that L is the limit of the series.