Video Transcript
Let us consider the function 𝑔 of
𝑥 is equal to 𝑥 divided by three minus 𝑥. Find a power series representation
for the function 𝑔 of 𝑥. Find the radius of convergence.
The question gives us a rational
function 𝑔 of 𝑥. It wants us to find the power
series representation of the function 𝑔 of 𝑥. And it wants us to then find the
radius of convergence for this power series. To do this, we recall a fact about
the sum of an infinite geometric series. The sum from 𝑛 equal zero to ∞ of
𝑎 multiplied by 𝑟 to the 𝑛th power is equal to 𝑎 divided by one minus 𝑟, when
the absolute value of our ratio 𝑟 is less than one.
So if we can write our rational
function 𝑔 of 𝑥 in the form 𝑎 divided by one minus 𝑟, then we can use this fact
about geometric series to write it as a power series with a radius of convergence
the absolute value of 𝑟 is less than one. We’ll start by noticing that we can
factor a coefficient of 𝑥 outside of our rational function. This gives us 𝑥 multiplied by one
over three minus 𝑥. We want our denominator to be in
the form one minus the ratio 𝑟. However, we have three minus 𝑥 as
our denominator. To have a denominator in the form
one minus 𝑟, we’re going to factor out a coefficient of three in our
denominator. This gives us three multiplied by
one minus 𝑥 divided by three. We can then simplify this by taking
the coefficient of three in our denominator and putting it outside of our rational
function.
If we then set 𝑎 equal to one and
𝑟 equal to 𝑥 divided by three, we can use our infinite geometric series to rewrite
our expression. Using our infinite geometric series
formula, we have that 𝑔 of 𝑥 is equal to 𝑥 divided by three multiplied by the sum
from 𝑛 equals zero to ∞ of one multiplied by 𝑥 over three to the 𝑛th power, when
the absolute value of our ratio 𝑥 divided by three is less than one. We can simplify this further.
First, we can remove the
multiplication by one. Next, we can take our coefficient
of 𝑥 divided by three inside of our sum to give us 𝑥 over three multiplied by 𝑥
over three to the 𝑛th power. However, we can simplify this
further to just be 𝑥 over three all raised to the power of 𝑛 plus one. This gives us that our function 𝑔
of 𝑥 is equal to the sum from 𝑛 equals zero to ∞ of 𝑥 over three to the power of
𝑛 plus one, when the absolute value of 𝑥 over three is less than one.
Next, the question wants us to find
the radius of convergence of this power series. We recall that we called 𝑟 the
radius of convergence of a power series if the power series converges when the
absolute value of 𝑥 is less than 𝑟 and diverges when the absolute value of 𝑥 is
greater than 𝑟. Unless the power series converges
for all values of 𝑥, in which case we call the radius of convergence ∞. We recall that for a geometric
series, the sum from 𝑛 equals zero to ∞ of 𝑎 multiplied by 𝑟 to the 𝑛th power
will diverge whenever the absolute value of our ratio 𝑟 is greater than one. In our case, our ratio 𝑟 is 𝑥
divided by three. So our power series converges when
the absolute value of 𝑥 over three is less than one and diverges when the absolute
value of 𝑥 over three is greater than one.
We can then rearrange both of these
expressions. The absolute value of 𝑥 over three
being less than one is the same as saying the absolute value of 𝑥 is less than
three. And if the absolute value of 𝑥
over three is greater than three, this is the same as saying the absolute value of
𝑥 is greater than three. So our power series converges when
the absolute value of 𝑥 is less than three and diverges when the absolute value of
𝑥 is greater than three. Therefore, our radius of
convergence 𝑟 is equal to three. And our power series for 𝑔 of 𝑥
will converge whenever the absolute value of 𝑥 is less than three.
