
Series 
Infinite series 
Alternating
series 
Alternating series test
or Leibnitz's
alternating series test 
Absolute
convergence 
Conditional
convergence 
Series
of positive terms 






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. 


Absolute
convergence 
An
infinite series 

is called absolutely convergent if the infinite series 

is convergent, i.e., 

the series
of absolute values of its
terms converge. 
Thus,
if a series is absolutely convergent, then both series, 



Example: Let
show that the series 

is
absolutely convergent. 

Solution: Given
series is absolutely
convergent since the series
of absolute values of its
terms 


converges
to 2, 



Therefore,
the series 

is
convergent and its sum is 2/3. 


Example: Let
show that the series 

is
absolutely convergent. 

Solution: Given
series is absolutely
convergent since the series
of absolute values of its
terms 

Therefore,
the series converges and its sum 



Conditional
convergence 
If
an infinite series 

is not convergent, it is called divergent. However, if it is convergent
but not 

absolutely convergent,
it is called conditionally convergent. 

Series
of positive terms 
Suppose
the series 

consists
of positive terms only. Then its partial sums 

form
an 

increasing sequence, therefore given series converges if and only if
the sequence of partial sums {s_{n}}
is bounded. 
We use this property for convergence tests of series with positive
terms. 

Example: The series of positive terms 

converges
since 

that
is, the sequence of partial sums is
bounded above, and the series converges to the number e. 
If
we calculate the partial sum of first 10 terms of the series,
s_{10
}= 2.718281
obtained
is the number e
to six
correct
decimal digits
what shows that the series
increases relatively fast. 








Calculus
contents B 



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