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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.


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