
Series 
Tests
for convergence 
Comparison
test 
Limit
comparison test 
Ratio
test 





Tests
for convergence 
Comparison
test 
A
series of positive terms converges if its terms, starting from the first
or one of the further terms, are not greater
than terms of the same index of a known convergent series of positive
terms. 

Suppose
that 

are
series of positive terms. Then 


(1)
If 

is
convergent and a_{n}
< b_{n} for all n,
then 

is
also convergent. 



(2)
If 

is
divergent and a_{n}
> b_{n} for all n,
then 

is
also divergent. 



Example: Let
prove that the series 

converges. 

Solution:
Since
n^{2} > (n

1)n
then 

In previous section was shown that the series 


converges
and its sum s_{
}= 1
therefore, by the comparison test, also converges the series 

and
the 

sum
of which 

as
Euler revealed (in 1736). 

Recall
that the harmonic series, i.e., the series of the
reciprocals of natural numbers diverges while the series of
the reciprocals of squares of natural numbers converges. The reason is,
the terms of the reciprocals of squares
decrease faster then the terms of the harmonic series. 

The
limit
comparison test 
Suppose 

are series of
positive
terms. If 


then
the series are either both convergent or both divergent. 

Example: Let
prove that the series 

converges. 

Solution: 



Since
both series are of positive terms, the limit is 1
> 0, and the series b_{n}
= 1/n^{2} is
convergent then, by the
limit
comparison test given series also converges. 

The
ratio test 
The test
for whether a series is absolutely convergent by testing the limit of the absolute
value of
the ratio of
successive
terms of the series is called the ratio test. 

(1)
If 

then
the series 

is
absolutely
convergent. 



(2)
If 

then
the series 

is
divergent.




(3)
If 

then
the test is inconclusive,
i.e.,
gives
no information. 



Example: Let
show that the series 

converges. 

Solution: Since, 

the series
converges. 


Example: Let
show that the series 

converges. 

Solution: 


thus,
the series
converges. 










Calculus
contents B 



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