Method of Undetermined Objects
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- Nested radicals Chapter 1: Introduction Nested radicals are something in the form of x1x2x3...n3n2n1\sqrt[n_1]{x_1\sqrt[n_2]{x_2\sqrt[n_3]{x_3...}}}n1x1n2x2n3x3... wher…
By Chong Jing Quan, Alex Federspiel
- Nested Radicals Chapter 2: Denesting 2-Nested and 3-Nested Radicals Part 1: Denest a 2-Nested Radical. Before I start, let's have some revision. Now, let's start denesting! Dene…
By Chong Jing Quan, Alex Federspiel
- Nested Radicals Chapter 3: Ramanujan's Radicals Ramanujan's Radicals are actually are radicals in the form of x+n+a=ax+(n+a)2+xa(x+n)+(n+a)2+(x+n)...a[x+(k−1)n]+(n+a)2+x[x+(n+1)…
By Chong Jing Quan, Alex Federspiel
- Nested Radicals Chapter 4: Nested Radicals of the type a+ba+ba+b...\sqrt{a+b\sqrt{a+b\sqrt{a+b\sqrt{...}}}}a+ba+ba+b... In this chapter we want to show that this type of rad…
By Chong Jing Quan, Alex Federspiel
- Nested Radicals Chapter 5: Relationship between Nested Radicals and Continued Fractions. Part 1: Introduction of continued fractions Continued fractions are those fractions in t…
By Chong Jing Quan, Alex Federspiel
- Nested Radicals Chapter 6: Identities derived from nested radicals Part 1: b+b1+4ω2=ωb2+ωb4+ωb8+...\frac{b+b\sqrt{1+4\omega}}{2}=\sqrt{\omega b^2+\sqrt{\omega b^4+\sqrt{\omega b…
By Chong Jing Quan, Bill Cooperman, Alex Federspiel
- Nested Radicals Chapter 7: Convergence of a n-th degree nested radical Recalling that in Chapter 4, we proved that a nested square root converges, but now, what if I ask you to …
By Chong Jing Quan, Alex Federspiel
- Nested Radicals Chapter 8: Existence of denesting in 2nd degree of nested radicals of the form α3−β3\sqrt{\sqrt[3]\alpha-\sqrt[3]\beta}3α−3β In this chapter, we are going to …
By Chong Jing Quan
- Lets say we want to find the sum and product of the roots of 5x2+3x+1=05x^2+3x+1=05x2+3x+1=0. Unfortunately the roots are complex. Well if r1r_1r1 and r2r_2r2 are the roots th…
By Mihir Borkar, Alex Federspiel