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p > 1 . □. The ratio test is particularly useful for series whose terms contain factorials or exponentials, where the ratio of terms simplifies 

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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. Practice: p-series.

P series test

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Ratio Test If for all n, n 0, then the following rules apply: Let L = lim (n -- > ) | a n+1 / a n |. If L < 1, then the series a n converges. If L > 1, then the series a n diverges. If L = 1, then the test in inconclusive.

∑. 1 n. (n2 + 1)2 converges .

The Integral Test; p-Series and The Harmonic Series. Day: 1 of 1 Then use the Integral Test to determine the convergence or divergence of the series. a. b.

= 1. 2 . Since the limit is less than 1, the Root Test says that the series converges absolutely.

P series test

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An alternative proof of the convergence of the p-series for p > 1 and divergence for 0 < p ≤ 1, using the integral test, is given in Example 12.44. Page 10. 68. 4.

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.

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