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Series
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Definition
►A
series is represented as the sum of a sequence
of infinite terms. That is, a series is a list of
numbers with addition operations between them.
 Ex. 1+1+1+1+………
► In
most cases of interest the terms of the
sequence are produced according to a certain rule,
such as by a formula, by an algorithm, by a
sequence of measurements, or even by randomly
generated numbers.
Infinite Series
►
The sum of an infinite series a0 + a1 + a2 + … + an, where
a0 + a1 + a2 + … + an are the terms of the sequence, is
the limit of the sequence of partial sums
 Sn = a0 + a1 + a2 + … + an , as n ∞ , if and only if the limit
exists.
In other words it is the sum of
S1 = a1
S2 = a1 + a2
S3 = a1 + a2+ a3
……………
Where S1, S2, S3, …, Sn are the sum of the terms in the sequence.
Infinite Series (cont.)
► More

formally, an infinite series is written as:
 Sn
n 0
where the elements in Sn are
real (or complex) numbers.
Convergence and Divergence
► If
the sequence of partial sums reaches a
definite value, the series is said to converge.
► On the other hand, if the sequence of
partial sums does not converge to a limit
(e.g., it oscillates or approaches +∞ or -∞),
the series is said to diverge.
Convergence and Divergence (cont.)
► More
formally, we say that the series
converges to M, or that the sum is M, if the
limit
K
Lim
Sn

K   n 0
exists and is equal to M. If there is no such
number, then the series is said to diverge.
Examples
► Convergent

n
Series
0
1
2
1 1 1 1
            ...  2

2 2 2
n 0  2 
► Divergent
Geometric Series
Series

1 1 1 1 1
       ...  

1 2 3 4
n 1  n 
Harmonic Series
Series Handouts and Links
► Series
Handout
► Sums and Series Handout
► Infinite Series Handout
► Solving Series Using Partial Fractions Handout
► Series
Quiz