THEOREM 12.4.2 THE RATIO TEST Letak be a series with positive terms and suppose that ak+1 λ. ak (i) If <1, then Σ a converges. (ii) If 1, then Σ a diverges. λ (iii) If = 1, then the test is inconclusive. The series may converge; it may diverge. THEOREM 12.4.1 THE ROOT TEST Letak be a series with nonnegative terms, and suppose that (ak) 1/k p. (i) If p < 1, then a converges. (ii) If p > 1, then a diverges. (iii) If p = 1, then the test is inconclusive. The series may converge; it may diverge.
THEOREM 12.4.2 THE RATIO TEST Letak be a series with positive terms and suppose that ak+1 λ. ak (i) If <1, then Σ a converges. (ii) If 1, then Σ a diverges. λ (iii) If = 1, then the test is inconclusive. The series may converge; it may diverge. THEOREM 12.4.1 THE ROOT TEST Letak be a series with nonnegative terms, and suppose that (ak) 1/k p. (i) If p < 1, then a converges. (ii) If p > 1, then a diverges. (iii) If p = 1, then the test is inconclusive. The series may converge; it may diverge.
Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+
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Explain the focus and reasons for establishment of 12.4.1(root test) and 12.4.2(ratio test)
Transcribed Image Text:THEOREM 12.4.2 THE RATIO TEST
Letak be a series with positive terms and suppose that
ak+1
λ.
ak
(i) If <1, then Σ a converges.
(ii) If 1, then Σ a diverges.
λ
(iii) If = 1, then the test is inconclusive. The series may converge; it may
diverge.
Transcribed Image Text:THEOREM 12.4.1 THE ROOT TEST
Letak be a series with nonnegative terms, and suppose that
(ak) 1/k
p.
(i) If p < 1, then
a converges.
(ii) If p > 1, then
a diverges.
(iii) If p = 1, then the test is inconclusive. The series may converge; it may
diverge.
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