### Video Transcript

Consider the series the sum from π
equals zero to β of π π, where π π is equal to π plus π factorial divided by
π to the πth power for some integers π and π, both strictly greater than
one. Calculate the limit as π
approaches β of the absolute value of π π plus one divided by π π. Hence, decide whether the series
converges or diverges.

The question gives us a series, the
infinite sum of π π, where it tells us that π π is equal to π plus π factorial
all divided by π to the πth power, where π and π are integers which are strictly
greater than one. The first part of this question
wants us to calculate the limit as π approaches β of the absolute value of the
ratio of successive terms in the sequence π π. The first thing we notice is,
instead of dividing by π π in our limit, we can multiply by the reciprocal of π
π. So we want to calculate the limit
as π approaches β of the absolute value of π π plus one multiplied by the
reciprocal of π π.

To calculate these values, weβll
use the definition of π π given to us in the question. We have that π π plus one is
equal to π plus one plus π factorial all divided by π to the power of π plus
one. And we multiply this by the
reciprocal of π π, which is π plus π factorial divided by π to the πth
power. We then take the reciprocal of π
π. This gives us the limit as π
approaches β of the absolute value of π plus one plus π factorial divided by π to
the power of π plus one multiplied by π to the πth power divided by π plus π
factorial. Weβre now ready to start
simplifying.

First, we notice that π plus one
plus π factorial is actually equal to π plus one plus π multiplied by π plus π
factorial. This means we can cancel the shared
factor of π plus π factorial in our numerator and our denominator. Similarly, we can cancel π of the
shared factors of π in our numerator and our denominator. This gives us the limit as π
approaches β of the absolute value of π plus one plus π all divided by π. We see that one does not vary
depending on the value of π. We also see that π does not vary
depending on the value of π. And we also see that π does not
vary depending on the value of π. However, our term of π is getting
larger and larger as π is approaching β. Therefore, the numerator of our
fraction is approaching β. However, the denominator remains
constant. Therefore, we can evaluate this
limit as β.

Finally, the question wants us to
use this to determine whether the series converges or diverges. To do this, we recall the ratio
test, which tells us if the limit as π approaches β of the absolute value of the
ratio of successive terms is less than one. Then the sum from π equals zero to
β of π π converges absolutely. And if the limit as π approaches β
of the absolute value of the ratio of successive terms is greater than one. Then the sum from π equals zero to
β of π π diverges. And we already calculated the limit
as π approaches β of the absolute value of the ratio of successive terms. And we showed it to be β. And in the case of the ratio test,
this also counts as this limit being greater than one. Therefore, we can conclude that our
series must diverge.

Therefore, for the sequence π π
is equal to π plus π factorial all divided by π to the πth power for some
integers π and π both greater than one. Since the absolute value of the
ratio of successive terms is equal to β, we can conclude by the ratio test that the
series must diverge.