Trigonometry Workbook For Dummies
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So-called p-series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A p-series is of the form

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(where p is a positive power). The p-series for p = 1 is called the harmonic series. Here it is:

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Although this grows very slowly — after 10,000 terms, the sum is only about 9.79! — the harmonic series in fact diverges to infinity.

By the way, this is called a harmonic series because the numbers in the series have something to do with the way a musical string like a guitar string vibrates — don’t ask. For history buffs, in the 6th century b.c., Pythagoras investigated the harmonic series and its connection to the musical notes of the lyre.

Here’s the convergence/divergence rule for p-series:

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As you can see from this rule, the harmonic series forms the convergence/divergence borderline for p-series. Any p-series with terms larger than the terms of the harmonic series diverges, and any p-series with terms smaller than the terms of the harmonic series converges.

The p-series for p = 2 is another common one:

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The p-series rule tells you that this series converges. It can be shown that the sum converges to

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But, unlike with the geometric series rule, the p-series rule only tells you whether or not a series converges, not what number it converges to.

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Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

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