Question Video: Determining Whether a Series Is Convergent or Divergent Using the 𝑝-Series Test Mathematics • Higher Education

Video Transcript

Use the 𝑝-series test to determine whether the series the sum from 𝑛 equals one to ∞ of 𝑛 to the power of 4.334 divided by 𝑛 to the power of 5.346 is divergent or convergent.

The question wants us to use the 𝑝-series test to determine the convergence or divergence of our series. Well, we can ask, what is the 𝑝-series test? The 𝑝-series test tells us that the 𝑝-series the sum from 𝑛 equals one to ∞ of 𝑛 divided by 𝑛 to the 𝑝th power is convergent if 𝑝 is greater than one and is divergent if 𝑝 is less than or equal to one. So to use the 𝑝-series test, we’re going to want to rewrite the series given to us in the question as a 𝑝-series.

To do this, we’re going to recall a fact about exponents. 𝑎 to the 𝑛th power divided by 𝑎 to the 𝑚th power is equal to 𝑎 to the power of 𝑛 minus 𝑚. Applying this to the summand of the series given to us in the question gives us that our series is equal to the sum from 𝑛 equals one to ∞ of 𝑛 to the power of 4.334 minus 5.346. Which we can simplify to give us the sum from 𝑛 equals one to ∞ of 𝑛 to the power of negative 1.012. We’re still not done yet. We need to write this as a 𝑝-series. And the summand of a 𝑝-series is of the form one divided by 𝑛 to the power of 𝑝.

So we’re going to use another exponent law to rewrite our series. 𝑛 to the power of negative 𝑚 is equal to one divided by 𝑛 to the 𝑚th power. This gives us that our series is equal to the sum from 𝑛 equals one to ∞ of one divided by 𝑛 to the power of 1.012. And we can see that this is now a 𝑝-series, where the value of 𝑝 is equal to 1.012. And our 𝑝-series test tells us that the 𝑝-series must be convergent if 𝑝 is greater than one. So because our value of 𝑝 is greater than one, the 𝑝-series test tells us that our series is convergent.

Therefore the 𝑝-series test tells us that the sum from 𝑛 equals one to ∞ of 𝑛 to the power of 4.334 divided by 𝑛 to the power of 5.346 is convergent because it is equivalent to a 𝑝-series where 𝑝 is greater than one.