27-Integral Test - Integral Test John E. Gilbert, Heather...

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Unformatted text preview: Integral Test John E. Gilbert, Heather Van Ligten, and Benni Goetz Some special infinite series like geometric series will play an important role, just as exponential and log functions did earlier in calculus. One such series is the Harmonic Series summationdisplay n = 1 1 n = 1 + 1 2 + 1 3 + 1 4 + . . . + 1 n + . . . . As the name suggests, it arises in the study of musical sounds. Does it converge or diverge? Well, 1 3 + 1 4 > 2 parenleftBig 1 4 parenrightBig = 1 2 , 1 5 + 1 6 + 1 7 + 1 8 > 4 parenleftBig 1 8 parenrightBig = 1 2 , . . . and so on; thus summationdisplay n = 1 1 n > 1 + 1 2 + parenleftBig 1 3 + 1 4 parenrightBig + parenleftBig 1 5 + 1 6 + 1 7 + 1 8 parenrightBig + . . . > 1 + 1 2 + 1 2 + 1 2 + . . . = since on the right we are now adding 1 2 infinitely many times. Thus the Harmonic Series diverges to ! But improper integrals can be made into a very useful test for infinite series such as the harmonic series. Let f ( x ) be a positive, decreasing function on [1 , ) . The graph of the sequence { f ( n ) } then appears as black dots at points ( n, f ( n )) on the graph of y = f ( x ) . To associate an area to the series n f ( n ) we draw green rectangles with base length 1 on the x-axis and one corner at these black dots as shown in extending either to the right or left; a rectangle with corner at ( n, f ( n )) will then have area f ( n ) . In the case of the harmonic series wed take f ( x ) = 1 x ....
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27-Integral Test - Integral Test John E. Gilbert, Heather...

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