Find the coefficient of $x^9$ in the power series of each of these functions.

Question

Find the coefficient of $x^9$ in the power series of each of these functions.

a) $(x^3 + x^5 + x^6)(x^3 + x^4)(x + x^2 + x^3 + x^4 +\cdots)$

b) $(x + x^4 + x^7 + x^{10} +\cdots )(x^2 + x^4 + x^6 + x^8 +\cdots)$

what should I do with this question? can you help? can you give any hint to me?

thanks.

Answer 1

Mathlove has given an excellent answer. I will only add this. In all computations you can drop any powers greater than 9. Also, if a term has a minimal power of $k$ then you don't need beyond $9-k$th power in the other terms combined.

So in a) the first term has minimum degree 3, the second also 3, so you don't need beyond $x^3$ in the third.

Answer 2

a)

$$(x^3 + x^5 + x^6)(x^3 + x^4)(x + x^2 + x^3 + x^4 +\cdots)$$ $$=x^3(x^3+x^4)(x + x^2 + x^3 + x^4 +\cdots)+x^5(x^3 + x^4)(x + x^2 + x^3 + x^4 +\cdots)$$$$+x^6(x^3 + x^4)(x + x^2 + x^3 + x^4 +\cdots)$$ $$=(x^6+x^7)(x + x^2 + x^3 + x^4 +\cdots)+(x^8+x^9)(x + x^2 + x^3 + x^4 +\cdots)$$$$+(x^9+x^{10})(x + x^2 + x^3 + x^4 +\cdots)$$ $$=(x^7+x^8+x^9+x^{10}+\cdots)+(x^8+x^9+x^{10}+\cdots)$$ $$+(x^9+x^{10}+\cdots)+(x^{10}+x^{11}+\cdots)$$ $$+(x^{10}+x^{11}+\cdots)+(x^{11}+x^{12}+\cdots).$$ So, the answer is $$1+1+1=3.$$

b) $$(x^1+x^4+x^7+x^{10}+\cdots)(x^2+x^4+x^6+x^8+\cdots)$$ $$=x^1(x^2+x^4+x^6+x^8+\cdots)+x^4(x^2+x^4+x^6+x^8+\cdots)$$ $$+x^7(x^2+x^4+x^6+x^8+\cdots)+x^{10}(x^2+x^4+x^6+x^8+\cdots)$$ $$=(x^3+x^5+x^7+x^9+\cdots)+(x^6+x^{10}+\cdots)$$ $$+(x^9+x^{11}+\cdots)+(x^{12}+\cdots)$$

So, the answer is $$1+1=2.$$

Answer 3

I focused on the second one since they contain infinite number of terms in geomatric progression.
The left sum is just $\dfrac{x^4} {(1 - x)}$ while the right sum is $\dfrac{x^2}{1 - x^2}$. Then their product is
$\dfrac{x^6}{(1 - x) (1 - x^2)}$.
Now, expand (Taylor or long division) $\dfrac{1}{(1 - x) (1 - x^2)}$ and stop at term $x^3$. You get your result of $2$.