What is a sequence in Calculus?

 Definition

A sequence is a list of numbers written in a specific order. The list can have a finite or an infinite number of terms. If it has a finite number of terms the sequence is finite. If the number of terms is infinite, the sequence is infinite. We will deal here with infinite sequence.

Notation

An infinite sequence can be written as follows:  

In this sequence  a₁ is the first term,  a₂ is the second term, ........., aₙ is the nth tern, aₙ₊₁ is the n+1 term.

This sequence can also be written as simply as {aₙ}.

Exercises

1) Write down the first few terms of the following sequences;

a)  {n + 1/n²} with n varying from 1 to infinite

b) (-1)^n+1/2ⁿ with n varying from 0 to infinite

Solutions

1) Let's have the first 5 terms of the sequence:

n =1 we have 1+1/(1)² = 2

n = 2 we have 2+1/(2)² = 3/4

n = 3 we have 3+1/(3)² = 4/9 

n = 4 we have 4+1/(4)² = 5/16

n = 5 we have 5+1/(5)² = 6/25

The sequence can be written as : {2, 3/4, 4/9, 5/16, 6/25, ..............}

2) Let's do the same thing for the second example:

n = 0 we have (-1)ٰ/2⁰ = -1

n = 1 we have (-1)²/2 = 1/2

n = 2 we have (-1)³/2² = -1/4

n = 3 we have (-1)⁴/2³ = 1/8

n = 4 we have (-1)⁵/2⁴ = -1/16

n = 5 we have (-1)⁶/2⁵ = 1/32

The sequence can be written as follows:

{-1, 1/2, -1/4, 1/8, -1/16, 1/32, .....}

Practice. Write the first few terms of the sequence