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?
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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.