
Series 
Infinite series 
The remainder
or tail
of the series 
Necessary
and sufficient
condition for the convergence of a series 
Necessary
condition for the convergence of a series 
The
nth
term test for divergence of series 
Properties
of series 
The
product of two series or the Cauchy product 
Geometric
series 
Pseries 





The remainder
or tail
of the series 
As
with sequences, the convergence of an infinite series 

only
depends on the behavior of the 

general
term of the series a_{n}
as n
increases to infinity, and not on any finite number of its initial
terms. 
Note
that, since 



the series 

converges
if and only if 

converges. 

Therefore,
to show that a series 

converges we can ignore any finite number of
terms 

at the 

beginning,
and just need to prove the convergence of the tail or remainder 

of
the series. 


The
difference between the sum s
of a convergent series a_{1}_{
}+
a_{2}_{
}+ a_{3}_{
}+
. . . + a_{n}_{
}+
. . .
and the n^{th}
partial sum s_{n}
is called the remainder (tail) r_{n}
of the series, i.e., 
r_{n
}= s 
s_{n}
= a_{n }_{+ 1 }+ a_{n }_{+ 2 }+
a_{n }_{+ 3 }+
. . .
or
s = s_{n}
_{
}+ r_{n}. 
Thus,
if a series 

converges
then the remainder r_{n}
= a_{n }_{+ 1 }+ a_{n }_{+ 2 }+
a_{n }_{+ 3 }+
. . . converges
too, 

that
is, since 

and
s
= s_{n}
+ r_{n}
then, 




Necessary
and sufficient
condition for the convergence of a series  Cauchy's convergence test 
Necessary
and sufficient
condition that the sequence of partial sums {s_{n}}
of a given series converges, and 
hence
the
series 

converges, is
that for given however small positive number e
it is possible to find 

an index n_{0} such
that 
s_{n
+ r} 
s_{n}
 < e
whenever
n
> n_{0}_{
}(e)
and r =
1, 2, 3,
. . . , 
or
expressed by terms of the series, if 

a_{n }_{+ 1 }+ a_{n }_{+ 2 }+
a_{n }_{+ 3 }+
. . . +
a_{n }_{+ r}_{ }

< e
whenever
n
> n_{0
}(e)
and r = 1, 2, 3,
. . . 
Therefore,
a series converges if the absolute value of the sum of any finite number
of sequential terms can become
arbitrary small by starting the addition from a term which is far
enough. 

Necessary
condition for the convergence of a series 
Hence, it is necessary
condition for the convergence of a series that its terms tend to
zero as n

increases to infinity,
that is 

So,
if this condition is not satisfied the series diverges. 

That
this condition is only necessary but not sufficient
condition for the convergence shows the harmonic
series for which 

as
was shown in previous section. 
Necessary
condition for the convergence of a series is usually used to
show that a series does not converge. 

The
nth
term test for divergence 

Note
that this is only a test for divergence. That is, if we can prove that
the sequence {a_{n}}
does not 
converge
to 0,
then the infinite series does not converge. 

Properties
of series 
If
given are two convergent series, 

then convergent is the series obtained by adding or subtracting their
same index terms, and its sum equals the
sum or the difference of their individual sums, i.e., 



The
product of two series or the Cauchy product 
If
given are two convergent series of positive terms, 

then the product 


denotes the
convergent series sum of which is equal to the product of the sums of
the given series. 

Geometric
series 
A
series, whose successive terms differ by a constant multiplier, is called a geometric series
and written as 

If
 x  < 1 ;
the nth
partial sum is 

Thus, the geometric series is convergent if
 x  < 1
and its sum is 

If
 x  > 1
then the geometric series
diverges. 

Example: Show
that the series 

converges. 

Solution:
Given
is the geometric series subsequent terms of which are multiplied by the
factor 1/2. 



Example: Let
prove
that the pure recurring decimal 0.333
... converges to 1/3.

Solution:
Given
decimal can be written as 


Example: Let
calculate the square of the convergent
geometric series


using
the Cauchy product shown above. 
Solution:



= 1 + 2x + 3x^{2}
+ 4x^{3} + · · · + (n + 1)x^{n} + · · · 
the
obtained series 

converges
for 0 < x^{}
<1. 



Pseries 
The
series 

converges
if
fixed constant
p
> 1 and diverges if p
< 1. 

By
grouping terms 


where
the number of terms in parentheses form the sequence 2,
4, 8, ... 2^{r1},
... such that 

therefore,
each value in parentheses is smaller than the corresponding term of the geometric
series 

Thus,
if p
> 1
then q
< 1,
the geometric series converges so that the given series is also
convergent. 
Euler
discovered and revealed sums of the series for
p
=
2m,
so for example 

If
p < 1
then n^{p}
< n or 1/n^{p}
> 1/n, therefore the
terms of the given series are not smaller than the terms 
of
the divergent harmonic series so, given series diverges. 









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