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Sometimes the series in this fact are called \(p\)-series and so this fact is sometimes called the \(p\)-series test. This fact follows directly from the Integral Test and a similar fact we saw in the Improper Integral section. This fact says that the integral, \[\int_{{\,k}}^{{\,\infty }}{{\frac{1}{{{x^p}}}\,dx}}\]

A commonly-used corollary of the integral test is the p-series test. Let k > 0 {\displaystyle k>0} . Then ∑ n = k ∞ ( 1 n p ) {\displaystyle \sum _{n=k}^{\infty }{\bigg (}{\frac {1}{n^{p}}}{\bigg )}} converges if p > 1 {\displaystyle p>1} . Integral Test and p-Series. The Integral Test. Consider a series S a n such that a n > 0 and a n > a n+1 We can plot the points (n,a n) on a graph and construct rectangles whose bases are of length 1 and whose heights are of length a n. If we can find a continuous function f(x) such that f(n) = a n P-Series Test As long as the a constant is below one and is being set to an exponent, p.

This test is actually a special case of the The Integral Test for Positive Series and is as follows: Theorem 1 (The p-Series Test): The special series is convergent if and divergent if . Proof of Theorem: Consider the series . Either this series converges or diverges. In general, a p-series follows the following form: p-series are useful because of the following theorem: The p-series is convergent if p > 1 and divergent otherwise. Unfortunately, there is no simple theorem to give us the sum of a p-series. For instance, the sum of the example series is . If p=1, we call the resulting series the harmonic series: By the above theorem, the harmonic series does not converge.

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## The N-Back Test is a test of Working Memory. Participants will be presented with and answer a series of simple, single digit math questions.

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(DAVE)— P-series is one of the most common types of series, and in this article, I go into detail. I explain what the p-series is and work through several examples on how to use the p-series test. That test is called the p-series test, which states simply that: If p > 1, then the series converges, If p ≤ 1, then the series diverges. Here are some examples of convergent series: Note the “p” value (the exponent to which n is raised) is greater than one, so we know by the test that these series will converge. A P-Series, also called a hyperharmonic series, is a modified harmonic series.

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Get access to all the courses and over 150 HD videos with your subscription. Monthly, Half-Yearly, and Yearly Plans Available. Get My Subscription Now. Not yet ready to subscribe? Take Calcworkshop for a spin with our FREE limits course (DAVE)— P-series is one of the most common types of series, and in this article, I go into detail. I explain what the p-series is and work through several examples on how to use the p-series test. That test is called the p-series test, which states simply that: If p > 1, then the series converges, If p ≤ 1, then the series diverges.