Cauchy Product of two Taylor Series

Question

I'm probably being a bit stupid here but I've been assigned this question and don't really know where to go with it.

Compute the first 5 terms of the Cauchy product of the Taylor Series for $(1-x)^{2/5}$ and $(1-x)^{3/5}$.

I've calculated the first five terms of the respective Taylor Series as

$$(1-x)^{2/5}= 1 - \frac{2x}{5} - \frac{3x^2}{25}- \frac{8x^3}{125}-\frac{26x^4}{625}$$

and

$$(1-x)^{3/5} = 1- \frac{3x}{5} - \frac{3x^2}{25} - \frac{7x^3}{125} - \frac{21x^4}{625}$$

I'd be really grateful if anyone could help me proceed with what to do next.

Answer 1

For safety, you can display the Cauchy product as an ordinary hand-made multiplication for numbers, truncating at order $5$. Here is an example of how to begin: