Answered: IV. Evaluate using root test, ratio…

IV. Evaluate using root test, ratio test and integral test (all three). State why a series is qualified or not for one of the tests Vn-1 п 8 п-0 In(n 9. п-0 2 10. Tn5n6 +

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

IV. Evaluate using root test, ratio test and integral test (all
three). State why a series is qualified or not for one of the
tests
Vn-1
п
8 п-0
In(n
9. п-0
2
10.
Tn5n6
+

Expert Solution

Step 1

(Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question specifying the one you’d like answered)

Root test is the test for the positive term series. It states that,

Leta, be a positive term series. Then the series
(a) Converges if lim(a )» <1.
(b) Diverges if lim(a,) >
n 1.
(c) Fails if lim(a)n = 1
Given
Vn-
Ση.
п
n-1
л -1
Let a
п
n-1
Vn-1
lim
2
lim (a
п
п-1
lim
"(/n+1)
n
1
lim 1
1
п
= 1
Hence, root test fails here.

Step 2

Ratio test for positive term series:

Leta, be a positive term series. Then the series
аp1
<1.
(a) Converges if lim
а,
(b) Diverges if lim
= 1
(c) Fails if lim
noa
Vn1
X
аp1
lim
п
= lim
n-o an
n+1
п -1
Vn1-
X
п
lim
n+1
1
1+
1
1
= lim
X
1
1
1
1+
no
n
п
= 1
Hence, ratio test also fails here

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