Using The P-series Test To Determine Convergence - Krista King Math

About Pricing Login Subscribe Risk Free Using the p-series test to determine convergence

What is the p-series test for convergence?

If we have a series ???a_n??? in the form

???a_n=\sum^{\infty}_{n=1}\frac{1}{n^p}???

then we can use the p-series test for convergence to say whether or not ???a_n??? will converge. The p-series test says that

???a_n??? will converge when ???p>1???

???a_n??? will diverge when ???p\le1???

The key is to make sure that the given series matches the format above for a p-series, and then to look at the value of ???p??? to determine convergence.

How to use the p-series test to determine convergence?

Take the course

Want to learn more about Calculus 2? I have a step-by-step course for that. :)

Learn More

Let’s do a couple more examples where we determine convergence or divergence using the p-series test

Example

Use the p-series test to say whether or not the series converges.

???\sum^{\infty}_{n=1}\frac{1}{\sqrt{n}}???

In order to use the p-series test, we need to make sure the format of the given series matches the format above for a p-series, so we’ll rewrite the given series as

???\sum^{\infty}_{n=1}\frac{1}{\sqrt{n}}=\sum^{\infty}_{n=1}\frac{1}{n^{\frac{1}{2}}}???

In this format, we can see that ???p=1/2???. The p-series test tells us that ???a_n??? diverges when ???p\le1???, so we can say that this series diverges.

Let’s try a second example.

Example

Use the p-series test to say whether or not the series converges.

???\sum_{n=1}^\infty\frac{1}{\sqrt[3]{n^4}}???

In order to use the p-series test, we need to make sure the format of the given series matches the format above for a p-series, so we’ll rewrite the given series as

???\sum_{n=1}^\infty\frac{1}{\sqrt[3]{n^4}}=\sum_{n=1}^\infty\frac{1}{(n^4)^\frac13}???

???\sum_{n=1}^\infty\frac{1}{n^\frac43}???

In this format, we can see that ???p=4/3???. The p-series test tells us that ???a_n??? converges when ???p>1???, so we can say that this series converges.

Get access to the complete Calculus 2 course

Get started Learn mathKrista KingMay 14, 2021math, learn online, online course, online math, calculus 2, calculus ii, p-series, p-series test for convergence, convergence or divergence, convergence tests, tests for convergence, sequences and series, sequences, series, infinite series Facebook0 Twitter LinkedIn0 Reddit Tumblr Pinterest0 0 Likes Previous

How to find the potential function of a conservative vector field

Learn mathKrista KingMay 15, 2021math, learn online, online course, online math, calculus iii, calculus 3, calc iii, calc 3, vector calculus, vector calc, potential function, conservative vector field, vector field that's conservative, open and simply-connected, scalar curl, line integrals, conservative vector fields, line integrals of conservative vector fields Next

Solving limits with conjugate method

Learn mathKrista KingMay 13, 2021math, learn online, online course, online math, calculus, calculus 1, calculus i, single variable calculus, single variable calc, limits, limits and continuity, conjugate method, solving limits, solving limits with conjugate method, conjugate method for limits, conjugate