# Question Video: Determining Whether a Series Is Convergent or Divergent Using the π-Series Test Mathematics • Higher Education

Use the π-series test to determine whether the series β_(π = 1) ^(β) (π^(4.334)/π^(5.346)) is divergent or convergent.

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### Video Transcript

Use the π-series test to determine whether the series the sum from π equals one to β of π to the power of 4.334 divided by π to the power of 5.346 is divergent or convergent.

The question wants us to use the π-series test to determine the convergence or divergence of our series. Well, we can ask, what is the π-series test? The π-series test tells us that the π-series the sum from π equals one to β of π divided by π to the πth power is convergent if π is greater than one and is divergent if π is less than or equal to one. So to use the π-series test, weβre going to want to rewrite the series given to us in the question as a π-series.

To do this, weβre going to recall a fact about exponents. π to the πth power divided by π to the πth power is equal to π to the power of π minus π. 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 π equals one to β of π to the power of 4.334 minus 5.346. Which we can simplify to give us the sum from π equals one to β of π to the power of negative 1.012. Weβre still not done yet. We need to write this as a π-series. And the summand of a π-series is of the form one divided by π to the power of π.

So weβre going to use another exponent law to rewrite our series. π to the power of negative π is equal to one divided by π to the πth power. This gives us that our series is equal to the sum from π equals one to β of one divided by π to the power of 1.012. And we can see that this is now a π-series, where the value of π is equal to 1.012. And our π-series test tells us that the π-series must be convergent if π is greater than one. So because our value of π is greater than one, the π-series test tells us that our series is convergent.

Therefore the π-series test tells us that the sum from π equals one to β of π to the power of 4.334 divided by π to the power of 5.346 is convergent because it is equivalent to a π-series where π is greater than one.