Video Transcript
Find a power series representation for the function 𝑓 of 𝑥 is equal to 𝑥 to the fourth power divided by 𝑥 plus seven.
The question is asking us to find a power series representation for our rational function 𝑓 of 𝑥. To write our function 𝑓 of 𝑥 as a power series, we want to show that 𝑓 of 𝑥 is equal to the sum from 𝑛 equals zero to ∞ of 𝑐 𝑛 multiplied by 𝑥 minus 𝑎 all raised to the 𝑛th power for some sequence 𝑐 𝑛 and some constant 𝑎. Since we’re asked to find the power series representation of a rational function, we can recall the following facts about geometric series.
The sum from 𝑛 equals zero to ∞ of 𝑏 times 𝑟 to the 𝑛th power is equal to 𝑏 divided by one minus 𝑟 if the absolute value of 𝑟 is less than one. And this series diverges if the absolute value of 𝑟 is greater than one. This gives us a method of turning a quotient into a power series. So, we want to write 𝑥 to the fourth power divided by 𝑥 plus seven in the form 𝑏 divided by one minus 𝑟. Since we have a single 𝑥 term in our numerator, we’ll just take 𝑥 to the fourth power outside of our fraction. We want to rewrite our denominator 𝑥 plus seven in the form one minus 𝑟. This means instead of seven, we want a constant one.
To achieve this, we’ll take a factor of seven outside of our denominator. This gives us 𝑥 to the fourth power multiplied by one divided by seven times 𝑥 over seven plus one. We can then take our factor of one-seventh outside of our fraction. This gives us 𝑥 to the fourth power divided by seven multiplied by one divided by one plus 𝑥 over seven. And we can see one divided by one plus 𝑥 over seven is of the form 𝑏 divided by one minus 𝑟. We have our initial value of 𝑏 equal to one and our ratio of successive terms, 𝑟, equal to negative 𝑥 divided by seven.
So, by using our method to find the infinite sum of a geometric series, we have our function 𝑓 of 𝑥 is equal to 𝑥 to the fourth power divided by seven multiplied by the sum from 𝑛 equals zero to ∞ of one times negative 𝑥 over seven all raised to the 𝑛th power. If the absolute value of negative 𝑥 over seven is less than one. We can simplify this. Multiplying by one does not change the value of our summand. And we can bring 𝑥 to the fourth power over seven inside of our sum. We can then also rewrite negative 𝑥 over seven as negative one times one over seven times 𝑥. This will then let us distribute the exponent over the parentheses.
Distributing the exponent over our parentheses gives us the sum from 𝑛 equals zero to ∞ of 𝑥 to the fourth power over seven multiplied by negative one to the 𝑛th power multiplied by one-seventh to the 𝑛th power multiplied by 𝑥 to the 𝑛th power. We have 𝑥 to the fourth power multiplied by 𝑥 to the 𝑛th power is 𝑥 to the power of 𝑛 plus four. We can rewrite one-seventh all raised to the 𝑛th power as one divided by seven to the 𝑛th power. We then see that one-seventh multiplied by one over seven to the 𝑛th power is one over seven to the power of 𝑛 plus one.
Rearranging our summand gives us that we found a power series representation for our function. 𝑓 of 𝑥 is equal to 𝑥 to the fourth power divided by 𝑥 plus seven is given by. The sum from 𝑛 equals zero to ∞ of negative one to the 𝑛th power divided by seven to the power of 𝑛 plus one multiplied by 𝑥 to the power of 𝑛 plus four.
