The Integral Test;Basic Comparison,Limit Comparison THEOREM 12.3.2 THE INTEGRAL TEST Iff is continuous,positive,and decreasing on [.),then converges iff f(x)dx converges f2）+…+fn) 7) (1)+…+fn-1) 3 4···n-1n 2 Figure 12.3.1 Main Meny 5
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The Integral Test;Basic Comparison,Limit Comparison Example 1 (The harmonic series) 111 (12.3.3) =1+2++4+… diverges. Main Meny C 007e
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Example 2 (The p-series) (12.3.4) =1+ 2p+3p+2 十· converges fp>1. Main Meny o
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Example 3 1 台kIn(k+) Main Menu 5
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The Integral Test;Basic Comparison,Limit Comparison In the absence of detailed indexing we cannot know definite limits,but we can be sure of the following: (I)if∑ak and∑bk converge,. then∑ak+bx converges: (2)for each o≠0 (12.3.5) ∑aak converges f ∑ak converges and,equivalently ∑aak diverges f ∑ak diverges. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. In the absence of detailed indexing, we cannot know definite limits, but we can be sure of the following: The Integral Test; Basic Comparison, Limit Comparison
The Integral Test;Basic Comparison,Limit Comparison THEOREM 12.3.6 THE BASIC COMPARISON THEOREM Suppose that >ag and >bg are series with nonnegative terms and ∑ak≤∑bi for all k sufficiently large. ()If∑br converges,.then∑ak converges (imIf∑ak diverges,then∑bdiverges. Main Meny o
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The Integral Test;Basic Comparison,Limit Comparison Example (a) 81 (b) 2g+城7 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example (a) (b) 3 1 2 1 k + 3 5 4 5 7 k k k + + The Integral Test; Basic Comparison, Limit Comparison
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Example 1 Main Meny☐ o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example 3 +1 1 k
Example 1 Z2n(k+6) Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example ln( + 6) 1 k