Sequences

Level 1 - Calculus topic page in Series And Approximations.

Principle

A sequence is an ordered list of numbers indexed by positive integers. Calculus studies whether sequence terms approach a limiting value as the index grows.

Notation

\(a_n\)

nth term of a sequence

\(n\)

positive integer index

\(\lim_{n\to\infty}a_n\)

limit of the sequence

The Core Method

To test convergence, analyse the expression for \(a_n\) as \(n\to\infty\).

Sequence convergence

\[a_n\to L\Longleftrightarrow \lim_{n\to\infty}a_n=L\]

For rational expressions in \(n\), divide by the highest power of \(n\).

Worked Cases

Question

Find \(\lim_{n\to\infty}\frac{3n+1}{2n-5}\).

Answer

Divide numerator and denominator by \(n\): \(\frac{3+1/n}{2-5/n}\). As \(n\to\infty\), \(1/n\to0\) and \(5/n\to0\). The limit is \(3/2\).

Examples

Question

Does \(a_n=(-1)^n\) converge?

Answer

The terms alternate between \(-1\) and \(1\). They do not approach a single number. Therefore the sequence diverges.

Arc Length

Series