Compute $$\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\cdots}}}$$
Simplify $\sqrt{5\sqrt{3}+6\sqrt{2}}$.
Simplify $\sqrt{12+2\sqrt{6}+2\sqrt{14}+2\sqrt{21}}$
Simplify $\sqrt{\sqrt[3]{9}+6\sqrt[3]{3}+9}$
Simplify $\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}$.
Solve $x^2 +6x - 4\sqrt{5}=0$.
Simplify $\sqrt{4+\sqrt[3]{81}+4\sqrt[3]{9}}$
Simplify $\sqrt{6+\sqrt[3]{81}+\sqrt[3]{9}}$.
For $n\ge 1$, let $d_n$ denote the length of the line segment connecting the two points where the line $y = x + n + 1$ intersects the parabola $8x^2 = y - \frac{1}{32}$ . Compute the sum $$\sum_{n=1}^{1000}\frac{1}{n\cdot d_n^2}$$
Find the length of the leading non-repeating block in the decimal expansion of $\frac{2004}{7\times 5^{2003}}$. For example the length of the leading non-repeating block of $\frac{5}{12}=0.41\overline{6}$ is $2$.
Find $x$ satisfying $x=1+\frac{1}{x+\frac{1}{x+\cdots}}$.
Write $\sqrt[3]{2+5\sqrt{3+2\sqrt{2}}}$ in the form of $a+b\sqrt{2}$ where $a$ and $b$ are integers.
Compute $$\sum_{n=1}^{\infty}\frac{2}{n^2 + 4n +3}$$
Compute $$\sum_{k=1}^{\infty}\frac{1}{k^2 + k}$$
Compute the value of $$\sum_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}$$
Let $(x^{2014} + x^{2016}+2)^{2015}=a_0 + a_1x+\cdots+a_nx^2$, Find the value of $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$
Find the length of the leading non-repeating block in the decimal expansion of $\frac{2017}{3\times 5^{2016}}$. For example, the length of the leading non-repeating block of $\frac{1}{6}=0.1\overline{6}$ is 1.
Let $\alpha_n$ and $\beta_n$ be two roots of equation $x^2+(2n+1)x+n^2=0$ where $n$ is a positive integer. Evaluate the following expression $$\frac{1}{(\alpha_3+1)(\beta_3+1)}+\frac{1}{(\alpha_4+1)(\beta_4+1)}+\cdots+\frac{1}{(\alpha_{20}+1)(\beta_{20}+1)}$$
Show that $$x+n=\sqrt{n^2 + x\sqrt{n^2+(x+n)\sqrt{n^2+(x+2n)\sqrt{\cdots}}}}$$
Find the value of $$\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
John uses the equation method to evaluate the following expression:$$S=1-1+1-1+1-\cdots$$ and get $$S=1-S \implies \boxed{S=\frac{1}{2}}$$
However, $S$ clearly cannot be a fraction. Can you point out what is wrong here?