

Series 
Infinite series 
The sequence of partial
sums 
The
sum of the series 
Convergence
of infinite series 
Divergence
of infinite series 
Convergent
and divergent series
examples 





Infinite series 
An
infinite series is the sum of infinite sequence of terms which we
denote 

That
is, given an infinite sequence of real numbers, a_{1},
a_{2}, a_{3}, . . . , a_{n},
. . .
all of the terms of which are added
together, where a_{n}
denotes the general term of the series. 

The
sequence of partial
sums 
Recall
that we've already dealt with infinite series in previous section when
representing real numbers as infinite
decimals. Thus, for example the real number 

so,
when adding up this infinite number of terms, we start by calculating
its partial sums 

Thus,
obtained is sequence {s_{n}}
of the partial sums, s_{1},
s_{2}, s_{3}, . . . , s_{n},
. . .
where s_{n}
denotes the sum of
the first n
terms of the series and is called the nth
partial sum. 
Therefore,
the
limit of the sequence {s_{n}}
as n
tends to infinity, i.e., 

equals the value of the real number 0.333
. . .
that is, equals the sum of
the infinite series. 

The
sum of the series 
An infinite series has a sum if the sequence of its partial sums
converge to a finite number s,
i.e., 

The sum of
an infinite series is the limit of its partial sums
as n
tends to infinity. 
Let
rewrite the above example using this notation, 

where, 



Convergence
of infinite series 
If
the
limit of the sequence of partial sums
exists
as a real number, then the series is convergent. 

Divergence
of infinite series 
If
the limit of the sequence of partial sums does not exist or tends
to +
oo
or 
oo, 

then, the series is
divergent or oscillates. 

Convergent
and divergent series
examples 
Example: Show
that the series 

diverges. 

Solution:
The given
infinite sum of natural numbers is called the arithmetic series. 
Using
the formula for the sum
of the arithmetic sequence, whose difference d
= 1,
we calculate the sum of the first n
terms of the series. 


Example: Let
show
that the series 

converges. 

Solution:
We
use the method of partial
fraction decomposition to rearrange given expression. 
Since
the denominator consists of linear factors then, we can write 

Therefore, 


Since
the second and the first term of any two successive parentheses cancel
then,
the sum of
the first n
terms of the series 

therefore the series converge, and its sum s
= 1. 

Example: Show
that the series 

converges. 

Solution:
Use the method of partial
fraction decomposition to rearrange given expression. 
Since
the
number of the coefficients used in the expansion relates to the degree
of the polynomial in the denominator,
and as it consists of linear factors some of which are repeated
then, 

Therefore, 


Since
the second and the first term of any two successive parentheses cancel
then, the sum
of
the first n
terms of the series 


therefore the series converge, and its sum s
= 1. 








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