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} .

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Test 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, \[\int_{{\,k}}^{{\,\infty }}{{\frac{1}{{{x^p}}}\,dx}}\] If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The LCT is a final resort test. If no other test is applicable or can work, we use the LCT. We create another series (bn) from the original function to compare n to. If bn converges, then an converges. If bn diverges, then an diverges.

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Test the series P 1 n=1 1 2+3 for convergence. Solution. Observe that 1 n2 + 3 < 1 n2 for every n 1. The series P 1 n=1 1 2 converges (p-series with p= 2 >1).

The p-series rule (or p-series test) states that if p>1, then the series converge. Otherwise, the series diverge. Notice that the series that we have is very similar to

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calculus real-analysis sequences-and-series convergence-divergence Share so we have an infinite series here one plus one over two to the fifth plus one over three to the fifth and we just keep on going forever we could write this as the sum from N equals one to infinity of one over N to the fifth power one over N to the fifth power and now you might recognize notice when N is equal to 1 this is 1 over 1 to the 5th that's that over there and we could keep on going now you might immediately recognize this as a p-series and a p-series … Thus the series and the integral converge or diverge together.
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This video tutorial explains the proof of P-Series Test with example. This video is useful for BSc students and those who are preparing for IIT JAM and CSIR

= 0 as n → с so the given series converges. Geometric Series Test (GST): The use of this test is straightforward. Typically, you choose bn to be either the p-series or the geometric series(basically the.

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Comparison tests for convergence. Worked example: p-series.

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