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.
