### 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 πth 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, weβre
going to want to rewrite the series given to us in the question as a π-series.

To do this, weβre going to recall a
fact about exponents. π to the πth power divided by π
to the πth 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. Weβre 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 weβre 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 πth 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.