Fraction of two series (help)

Question

I've ran across the following expression:

$$ \dfrac{a_1 x + 2a_2 x^2 + 3a_3 x^3 +\ ...}{1 + a_1 x + a_2 x^2 + a_3 x^3 + \ ...} $$

Now one is supposed to be able to write this fraction of series as:

$$a_1 x + 2 \left( a_2 - 1/2 a^2_1 \right)x^2 + 3 \left( a_3 - a_1 a_2 + 1/3 a^3_1 \right) x^3 + \ ...$$

Any hints on how to do this ?

Hint: try writing the result as a power series with unknown coefficients, say $b_0 + b_1 x + b_2 x^2 + \cdots$. Then multiply up and compare coefficients of powers of $x$. — Chris Lewis, Jul 24 '23 at 12:04
Long division, perhaps? (If you don’t need too many terms…) — David Quinn, Jul 24 '23 at 13:48

score 1 · Answer 1 · answered Jul 30 '23 at 06:57

Assuming that the problem is $$f(x)=\frac{\sum_{n=1}^p n\,a_n\ x^n }{1+\sum_{n=1}^pa_n\,x^n }$$ rewrite it as $$\frac{f(x)}x=\frac{\sum_{n=1}^p n\,a_n\ x^{n-1} }{1+\sum_{n=1}^pa_n\,x^n }$$ Integrate $$\int \frac{f(x)}x\,dx=\log\Bigg(1+\sum_{n=1}^p a_n\,x^n \Bigg)$$ Now, use $$\log(1+t)=\sum_{k=1}^\infty (-1)^{k+1} \, \frac {t^k} k$$ Make $$t=\sum_{n=1}^p a_n\,x^n$$ and use the binomial expansion.

When done, differentiate and multiply by $x$ to have the series expansion of $f(x)$.

Fraction of two series (help)

1 Answers1