Consider these two functions $$F(x) = \sum_{n=0}^{\infty} a_nx^n = \frac{1-5x}{1-4x+3x^2}$$

and $$G(x) = \sum_{n=0}^{\infty} a_nx^n = \frac{3+3x}{1+x-2x^2}$$

I can see from this post that the general strategy is to expand the polynomial, but the second method given, for a fraction, is a trick that doesn't work in this case.

How can I solve these two problems?


Let's look at one and the other works the same way. Since we can factorize the denominator we can write it as partial fractions.

$$F(x)=\frac{1-5x}{3x^2-4x+1}=\frac{2}{1-x}-\frac1{1-3x}=\left(2\sum_{i=0}^{\infty}x^i\right)-\left(\sum_{i=0}^{\infty}(3x)^i\right) \\ \implies F(x)=\sum_{i=0}^{\infty}(2-3^i)x^i$$

Note: Sum of geometric series $1+x+x^2+\cdots = \frac{1}{1-x}$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.