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The p-series test. A test exists to describe the convergence of all p-series. 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:
30 cze 2023 · Learn how to use the p-series test to determine the convergence or divergence of a series of the form ∑ (1/nᵖ). Explore the historical significance, properties, and generalizations of this test in mathematical analysis.
30 lip 2024 · Learn how to use the p-series test to determine the convergence or divergence of infinite series of the form ∑1/np. See examples, conditions, comparisons with other tests, and practice problems.
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} .
Learn what a p-series is, how to test its convergence using the integral test, and see some examples. A p-series is a series of the form , where p is a positive real number.
Using the p-Series test, determine whether the series $\sum_{n=1}^{\infty} \frac{1}{n^5}$ is convergent or divergent. By the p-Series test we note that $p = 5 > 1$ and therefore the series is convergent.
18 paź 2023 · Learn about the p-series, a benchmark series that depends on a parameter p, and how to test its convergence or divergence. See examples, comparisons, integrals, and graphs of p-series and related functions.
