### Video Transcript

What condition must be true of the
common ratio, π, in a geometric sequence if the sum of infinitely many terms can be
found?

In this question, we are asked to
find the condition on the common ratio π that guarantees that the limit of the
infinite geometric series will exist.

There are many ways to answer this
question. The easiest is to recall that the
infinite sum of a geometric sequence with initial term π and constant ratio π
whose absolute value is less than one is π over one minus π. We can then recall that a geometric
sequence with common ratio whose size is greater than or equal to one does not
necessarily converge. So we can say that the absolute
value of π must be less than one. It is useful to be able to recall
properties and theorems in this way. However, it is also important to
understand why these results hold true. So, letβs analyze this question in
more detail.

First, we can start by recalling
that a geometric sequence is one in which we find the next term in the sequence by
multiplying by a constant value of π. In other words, the ratio of
successive terms in the sequence remains constant. Therefore, if the initial term in
the geometric sequence is π and the common ratio is π, we obtain a sequence π,
ππ, ππ squared, ππ cubed, and so on.

Before we consider the sum of these
terms, letβs start by considering the trivial cases of these types of sequences. First, if π is zero, then every
term is zero regardless of the value of π. This will not yield any useful
information about the conditions of π that make the infinite sum exist, so we can
ignore this case. Similarly, if π is zero, then
every term after the first is zero. So the infinite sum is equal to the
first term π. Thus, we know that any geometric
sequence with π equals zero has an infinite sum that converges.

We are now ready to consider the
general case of a sum of the terms of a geometric sequence. Letβs call the sum of the first π
terms π sub π, and we can write the sum out as shown. We can now find an expression for
π times π sub π by multiplying every term by π. We obtain the following
expression. We can use this to find an
expression for the difference between π sub π and π times π sub π. If we subtract the second equation
from the first equation, we can note that all of the terms except π and ππ to the
πth power cancel. This leaves us with the equation π
sub π minus ππ sub π equals π minus ππ to the πth power.

We can rewrite this equation by
taking out the shared factor of π sub π on the left-hand side of the equation and
the shared factor of π on the right-hand side of the equation to get the
following. We can then divide the equation
through by one minus π to obtain that π sub π equals π times one minus π to the
πth power over one minus π. We can note that this expression
for the sum of the first π terms is not valid when π equals one, since we cannot
divide by zero. Thus, we will have to consider this
case separately.

We want to consider what happens
when we take the sum of an infinite number of terms. So, we want the value of π to
approach β. We can see that there is only one
term in this expression that changes as the value of π changes. We know that if the size of π is
less than one, then π to the πth power will get smaller in size as π
increases. Similarly, if the size of π is
greater than one, then the size of π to the πth power is unbounded as π
increases. This shows two of the possible
cases. When the size of π is less than
one, we can remove the term π to the πth power in our numerator to obtain our
infinite sum result. However, if the size of π is
greater than one, then the size of the terms gets larger and larger, so the sum does
not converge.

There are two cases we have not yet
considered. First, when π equals one, every
term in the geometric sequence is equal to π. If we then add the first π terms
of the sequence together, then we get π times π. This will only converge as π gets
larger if π is zero. Thus, it does not converge in
general. Second, we need to consider what
happens when π equals negative one. We see that the terms of the
sequence are the same size but switching in signs.

To see why the sum of infinitely
many terms of this sequence does not converge in general, we can consider each sum
of π terms. This gives us a sequence called the
sequence of partial sums. In this case, the sequence is π,
zero, π, zero, and this sequence continues indefinitely. This only converges if π is
zero. So we can say that the sum of an
infinite number of terms of a geometric sequence is guaranteed to converge if the
absolute value of π is less than one.