The limit comparison test states: an bn Suppose [an, [bn are series of positive terms and that L = lim % € (0, ∞). Then the series have the same convergence status (both converge or both diverge to ∞). 1 n² n (a) Use the limit comparison test with b₁ = to show that the series Σ In (1+1) converges. (Hint: Recall that e = lim (1 + ½)”) (b) Prove the limit comparison test. L (Hint: first show that ½ < a <31 for large n) an 2 bn 2 (c) What can you say about the series Σan and Σ bn if L = 0 or L = ∞? Explain.
The limit comparison test states: an bn Suppose [an, [bn are series of positive terms and that L = lim % € (0, ∞). Then the series have the same convergence status (both converge or both diverge to ∞). 1 n² n (a) Use the limit comparison test with b₁ = to show that the series Σ In (1+1) converges. (Hint: Recall that e = lim (1 + ½)”) (b) Prove the limit comparison test. L (Hint: first show that ½ < a <31 for large n) an 2 bn 2 (c) What can you say about the series Σan and Σ bn if L = 0 or L = ∞? Explain.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.4: Improper Integrals
Problem 37E
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![The limit comparison test states:
an
bn
Suppose [an, [bn are series of positive terms and that L = lim % € (0, ∞). Then the
series have the same convergence status (both converge or both diverge to ∞).
1
n²
n
(a) Use the limit comparison test with b₁ = to show that the series Σ In (1+1) converges.
(Hint: Recall that e = lim (1 + ½)”)
(b) Prove the limit comparison test.
L
(Hint: first show that ½ < a <31 for large n)
an
2 bn 2
(c) What can you say about the series Σan and Σ bn if L = 0 or L = ∞? Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ada9a20-cfc4-4295-96e5-fd4aa8bf1cec%2F9285bdba-5748-4c76-8fe9-c9865a409b9f%2F4igze8_processed.png&w=3840&q=75)
Transcribed Image Text:The limit comparison test states:
an
bn
Suppose [an, [bn are series of positive terms and that L = lim % € (0, ∞). Then the
series have the same convergence status (both converge or both diverge to ∞).
1
n²
n
(a) Use the limit comparison test with b₁ = to show that the series Σ In (1+1) converges.
(Hint: Recall that e = lim (1 + ½)”)
(b) Prove the limit comparison test.
L
(Hint: first show that ½ < a <31 for large n)
an
2 bn 2
(c) What can you say about the series Σan and Σ bn if L = 0 or L = ∞? Explain.
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