Find the first non-zero terms of Power Series

Question

Find the first five non-zero terms of power series representation centered at �=0 for the function below. f(x)= x^3/(1+5x)

Answer: f(x)= ?

What is the interval of convergence?

Answer (in interval notation):

Answer 1

Let's think about a slightly simpler version of the problem:

f(x) = x^3 / (1+x).

You can view f(x) = x^3 * (1/(1+x))

Power series for (1/(1+x)) seems a bit familiar to (1 / (1-x)).

1/(1-x) is written in the form a₁ / (1-r) which is the formula for the sum of an infinite geometric series with first term as a₁ and r as the common ratio.

In 1/(1-x), the first term of the geometric series is 1 and the common ratio is x, so the power series would be 1 + x + x^2 + ....

However, we have 1/(1+x) instead, which is equivalent to 1/(1 - (-x)) so the first term of the geometric series would be 1, but the common ratio is -x, so the power series is 1 - x + x^2 - x^3 + x^4 + ....

Now that we know how to write power series for 1/(1-x), we see that all we have to do is multiply each of the terms by x^3 to get the power series for x^3 * 1/(1-x).

Ok, I understand, but how to apply this to your original question?

Notice that x^3/(1+5x) = x^3 * (1/(1+5x)) and 1/(1+5x) = 1/(1 - (-5x)). Geometric series with common ratio -5x and first term 1. Power series for that part is 1 - 5x + 25x^2 - 125x^3 + ....

Then, multiply each of those terms by x^3, so the power series is:

x^3 - 5x^4 + 25x^5 - 125x^6 + 625x^7 + ... = ∑^infinity_j=1 ((x³) * (-5x)^j).

Now that you know the series representation, you can try applying a convergence test such as the Ratio Test to determine the interval of convergence! Feel free to let me know if you'd like any followups