
Series 
Alternating
series 
Alternating series test
or Leibnitz's
alternating series test 






Alternating
series 
A
series the terms of which are alternately positive and negative is
called the alternating series. 
Alternating series test
or Leibnitz's
alternating series test 
An
alternating series converge if the absolute values of its terms decrease
monotonically to zero as n
tends to
infinity. 
Given
an alternating series a_{1}

a_{2} + a_{3} 
+
·
·
·
+
(1)^{n

}^{1}
a_{n} +
·
·
· 
suppose
a_{1}
> a_{2} > a_{3} > ·
·
·
> a_{n}
> a_{n
+ }_{1} > ·
·
·
> 0
and 


The
sequence of the partial sums s_{n}
of the series alternate approaching the same limit s.
The terms of s_{2n
}of the
even
indices,
increase while the terms of s_{2n
+ 1 }of the odd indices,
decrease as is shown on the number line
below. Every term of the sequence of an even index is smaller than every
term of an odd index therefore, s_{2n}
< s
< s_{2n
+ 1} 

and 




Example: Let
show that the series 

converges. 

Solution: The
given alternating series converges since the sequence of absolute values of its
terms 


that
is, 



decrease to zero as
n
tends to infinity, or 

The
series converges to ln
2 or s
= ln 2. 

Note
that the given logarithmic series is not absolutely convergent since 

obtained
harmonic series diverges. 
Therefore,
the series 

is
said to be conditionally convergent. 


Example: Let
show that the series 

converges. 

Solution: The
given alternating series converges since the sequence of absolute values of its
terms 


that
is, 



decrease to zero as
n
tends to infinity, or 

As
Leibnitz proved, the
series converges to p/4. 










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