This PPT used in first sem of engineering in Calculus subjects to make seminar. This ppt is also useful for all the engineering students in MATH 1 subject.

1. PANCHAL DHRUMIL 17BECEG104
WELCOME

2. Ratio Test
 If ∑an is a positive-term series and
then
i. ∑an is convergent if L<1
ii. ∑an is divergent if L>1
iii. The test is inconclusive if L=1
1
lim n
n
n
a
L
a




3. Example 1


1 !
1
k k
 Use the ratio test to determine whether the following
series converge or diverge.
(a)
converges
kkk
kk
ku
u
k
k
k
10
1
1
!)1(
!
1
!
)!1(
1
lim 1






 



4. Example 2


1 2k
k
k
 Use the ratio test to determine whether the following
series converge or diverge.
(b)
converges
k
k
k
k
k
k
u
u
k
kk
k
k
k
k
1
2
1
2
1
22
)1(22
2
1
lim 1
1







 




5. Example 3
 Use the ratio test to determine whether the
following series converge or diverge.
(c)
 
diverges
e
k
k
k
k
k
k
u
u k
k
kk
k
k
k
k
k
k
11lim
)1(!
)!1(
)1(
lim 1
1
1













1 !k
k
k
k

6. Example 4
 Use the ratio test to determine whether the
following series converge or diverge.
(d) 

1 4
)!2(
k
k
k
diverges
kk
k
k
u
u k
k
k
k
k
1
4
)12)(22(
)!2(
4
4
)]!1(2[
lim 1
1




 




7. Root Test
 Let ∑Un be a non negative series, and
assume that lim n→∞n√Un=L (possibly
∞)
a. If 0<=L<1 , then ∑Un converges .
b. If L>1 , then ∑ Un diverges.
If L=1 then from this test alone we can
not draw any conclusion about the
convergence or divergence of ∑Un.

9. Example 2:
2
1 2n
n
n


2
2
n
n
n 2
2
n
n

2
lim
2
n
n
n

2
lim n
n
n

 
2
lim n
n
n


2
1 1
1
2
  it converges
?

10. Thank you very much Everyone