# Definition and examples.

Definition 9.1.1   Let be given a sequence of real numbers. The expression

is called the infinite series with as its general term.

Definition 9.1.2   The sequence whose general term is is called the sequence of partial sums of the given series.

Definition 9.1.3   The series is convergent if its sequence of partial sums is convergent. Otherwise it is divergent.

if the sequence of partial sums converges, its limit is called the sum of the series.

For the definition of the convergence of a sequence, v.s. Def. 6.1.

Proposition 9.1.4 (Geometric Series)   A geometric series is a series , where is a geometric sequence. The ratio of the sequence is also called the ratio of the geometric series.

Denote the first term of the geometric series and by its ratio; according to Prop. 4.2, we have: . thus:

• If , the series is convergent and its sum is equal to .
• If , the series is divergent.

Example 9.1.5   Let , i.e. . The number x is the sum of a geometric series, whose first term is equal to and whose ratio is equal to . Thus:

Example 9.1.6 (A Telescoping Series)   Consider the series whose general term is . we have:

Hence:

It follows that the series . is convergent and its sum is equal to 1.

Proposition 9.1.7   We consider the series . If it is convergent, then .

Example 9.1.8   Once again, consider the telescopic series from Example 1.6. We have: .

Take the series:

Despite the fact that the general term of the series has a limit equal to 0, this series is divergent (it has an infinite sum).

Example 9.1.9 (The harmonic series)   Take the series .

We prove that this series is divergent by grouping the terms in the following way:

Noah Dana-Picard 2007-12-28