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4. (a) Prove that $\dfrac{1-x^2}{1+x^3}=\dfrac{1}{1+\frac{x^2}{1-x}}$. (b) By expanding each side of the identity in (a) as a power series, and considering the coefficient of $x^N$, prove …

My textbook has a whole bunch of exercises on finding some coefficient inside a formal power series. Unfortunately, there aren't any examples on how to do so, especially since many of the series expan …

Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies $$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$ Determine a recurrence relation that ${a_n}$ satisfies. I mult …

Express $$\sum^{20}_{i=2}f(x)^i$$where $$f(x)=\sum_{i\geq 1}2^{i-1}x^{3i}$$ as a fraction of polynomials $p(x)/q(x)$ and simplify as much as possible. Hmm. How to do it? Wolfram is really stupid on …

Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies $$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$ The denominator can be factored into $(1-2x)(1+x)^2$. Using res …

Using Mathematical induction on $k$, prove that for any integer $k\geq 1$, $$(1-x)^{-k}=\sum_{n\geq 0}\binom{n+k-1}{k-1}x^n$$ How should I proceed? The tutorial teacher attempted this question and f …