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 $p\le1$

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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.

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Let’s try a second example.

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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.

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.

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