I know the higher-math answer to this question, but I'm asking how on Earth to explain it to a bright high school student. Here's the question, paraphrased: “I can see that $\left(1+x+x^{2}+x^{3}+x^{4}+....\right)\left(1-x\right)=1$ because the terms with non-zero powers of $x$ all cancel out. ...

I need help with these 3 equations. 1) $(1+y^2)dx+xydy=0$ 2) $y''-2y+2y=0$, $y(0)=4$, $y'(0)=2$ 3) $y'-xy=x^3\cdot y^3$

In the text "Function Theory of One Complex Variable" Third Edition by Robert E.Greene and Steven G.Krantz. I'm having trouble proving the following case of the root test in $(1.)$ $(1.)$ $\text{Lemma 3.2.6 (The root test)}:$ The radius of convergence of the power series $\sum_{} a_{k}(z-P)^{...

In the text "Real and Complex Integration Problems" I'm having trouble verifying my approach to proving following conjecture in $(2.).$ $(0.)$ $(0.1)$ $$K_{n}(\theta) = \sum(1 - \frac{|j|}{n+1})$$ $(1.)$ Show that for any continuous $2 \pi$ periodic function $f$ on $\mathbb{R}$ one has: $$ K...

In the text "Real and Complex Integration Problems" I'm having trouble verifying my approach in proving the following conjecture in $(2.).$ $(0.)$ $(0.1)$ $$K_{n}(\theta) = \sum(1 - \frac{|j|}{n+1})$$ $(1.)$ Show that for any continuous $2 \pi$ periodic function $f$ on $\mathbb{R}$ one has: ...

In the text "Function Theory of One Complex Variable" Third Edition by Robert E.Greene and Steven G.Krantz. I'm having trouble proving the following case of the root test in $(1.)$ $(1.)$ $\text{Lemma 3.2.6 (The root test)}:$ The radius of convergence of the power series $\sum_{} a_{k}(z-P)^{...