
Series 
Tests
for convergence 
Ratio
test 
Root
test or Cauchy's root test 





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: Examine
whether the series 

converges
or diverges. 

Solution: Since, 



the
series converges if 0
< x < 1. 

Example: Determine
whether the series 

converges
or diverges. 

Solution: Since, 


therefore L
< 1 and the series converges. 
If
the given series has a factorial in the nth
term expression, the ratio test
should be used. 

Example: Let
show that the logarithmic series 

converges. 

Solution: 


thus, the series
converges
for x < 1.
The sum of the series 



Root
test or Cauchy's root test 
The test
for whether a series 

of
positive terms is absolutely
convergent by considering; 


(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: Determine
whether the series 

converges
or diverges. 

Solution: The
given geometric series converges since 




Example: Let
show that the series 

converges. 

Solution: The
series converges since 




Example: Test
the series 

for
convergence. 

Solution: The
series converges since 



We
use
the root test when the entire nth
term expression is raised to some power of n. 








Calculus
contents B 



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