SeriesSeries

Brought to you byBrought to you by

Tutorial Services – The Math Tutorial Services – The Math CenterCenter

DefinitionDefinition

► A series is represented as the sum of a A series is represented as the sum of a sequence of infinite terms. That is, a series is a sequence of infinite terms. That is, a series is a list of numbers with addition operations list of numbers with addition operations between them.between them. Ex. 1+1+1+1+………Ex. 1+1+1+1+………

► In most cases of interest the terms of the In most cases of interest the terms of the sequence are produced according to a certain sequence are produced according to a certain rule, such as by a formula, by an algorithm, by rule, such as by a formula, by an algorithm, by a sequence of measurements, or even by a sequence of measurements, or even by randomly generated numbers.randomly generated numbers.

Infinite SeriesInfinite Series

► The sum of an infinite series aThe sum of an infinite series a00 + a + a11 + a + a22 + … + a + … + ann, , where awhere a00 + a + a11 + a + a22 + … + a + … + an n are the terms of the are the terms of the sequence, is the limit of the sequence of partial sequence, is the limit of the sequence of partial sumssums SSnn = a = a00 + a + a11 + a + a22 + … + a + … + an , n , as n as n ∞ , if and only if the limit ∞ , if and only if the limit

exists.exists.In other words it is the sum ofIn other words it is the sum of

SS11 = a = a11

SS22 = a = a11 + a + a22

SS33 = a = a11 + a + a22+ a+ a33

…………………………

Where SWhere S11, S, S22, S, S33, …, S, …, Snn are the sum of the terms in the are the sum of the terms in the sequence.sequence.

Infinite Series (cont.)Infinite Series (cont.)

►More formally, an infinite series is More formally, an infinite series is written as:written as:

where the elements in Sn where the elements in Sn are are

real (or complex) numbers.real (or complex) numbers.

0n

Sn

Convergence and DivergenceConvergence and Divergence

► If the sequence of partial sums If the sequence of partial sums reaches a definite value, the series is reaches a definite value, the series is said to converge.said to converge.

►On the other hand, if the sequence of On the other hand, if the sequence of partial sums does not converge to a partial sums does not converge to a limit (e.g., it oscillates or approaches limit (e.g., it oscillates or approaches ++∞ or -∞), the series is said to diverge.∞ or -∞), the series is said to diverge.

Convergence and Divergence Convergence and Divergence (cont.)(cont.)

►More formally, we say that the series More formally, we say that the series converges to M, or that the sum is M, if converges to M, or that the sum is M, if the limitthe limit

exists and is equal to M. If there is no exists and is equal to M. If there is no such number, then the series is said to such number, then the series is said to diverge.diverge.

K

n

SnK

Lim

0

ExamplesExamples

► Convergent SeriesConvergent Series

►Divergent SeriesDivergent Series

2...2

1

2

1

2

1

2

1210

0

n

n

Geometric Series

...4

1

3

1

2

1

1

11

1n nHarmonic Series

Series Handouts and LinksSeries Handouts and Links►Series Handout►Sums and Series HandoutSums and Series Handout► Infinite Series HandoutInfinite Series Handout►Solving Series Using Partial Fractions Solving Series Using Partial Fractions

HandoutHandout► Series QuizSeries Quiz