Question

State the definitions of convergent and divergent series.

Short answer

A convergent series is a series whose sequence of partial sums tends to a finite limit. Conversely, a divergent series is a series whose sequence of partial sums does not tend to a finite limit.

Step-by-step solution

Define Convergent Series

A series \( \sum a_n \) is said to be convergent if the sequence of its partial sums \( S_n = a_1+a_2+a_3+...+a_n \) tends to a finite limit as \( n \rightarrow \infty \). That is, if there exists a finite number 'l', such that for every real number \( \epsilon >0 \), there exists a positive integer \( N \) such that for all \( n \geq N\), \( |S_n-l|< \epsilon \), then the series \( \sum a_n \) is said to be convergent series.

Define Divergent Series

If a series \( \sum a_n \) is not convergent, i.e., the sequence of its partial sums \( S_n = a_1+a_2+a_3+...+a_n \) does not tend to a finite limit as \( n \rightarrow \infty \), then the series is said to be divergent.