Series

Level 1 - Calculus topic page in Series And Approximations.

Principle

A series is the sum of sequence terms. Its convergence depends on whether the partial sums approach a finite limit.

\(\sum_{n=1}^{\infty}a_n\)

infinite series

\(S_N\)

Nth partial sum

\(a_n\)

nth term of the series

Define partial sums and take their limit.

Partial sum

\[S_N=a_1+a_2+\cdots+a_N\]

Series convergence

\[\sum_{n=1}^{\infty}a_n\text{ converges }\Longleftrightarrow \lim_{N\to\infty}S_N\text{ exists finitely}\]

If \(a_n\) does not approach zero, the series diverges.

Question

Find \(\sum_{n=0}^{\infty}\left(\frac12\right)^n\).

Answer

This is a geometric series with first term \(1\) and ratio \(r=1/2\). Since \(|r|<1\), it converges to \(\frac{1}{1-r}=\frac{1}{1-1/2}=2\).

Question

Why does \(\sum_{n=1}^{\infty}\frac{n}{n+1}\) diverge?

Answer

The terms satisfy \(\frac{n}{n+1}\to1\), not \(0\). Since a convergent series must have terms tending to zero, the series diverges by the term test.