How to solve $10\sqrt{10\sqrt[3]{10\sqrt[4]{10...}}}$?
I tried to solve this problem by letting $x=10\sqrt{10\sqrt[3]{10\sqrt[4]{10...}}}$ to observe the pattern.
Based on the pattern, the result is
$\dfrac{x^{n!}}{10^{((((1)(2)+1)4+1)...n+1)}}$ where $n$ is a positive integer approaching infinity.
This is where I got stuck.
On
We can rewrite your expression as: $$S=10\cdot(10\sqrt{...})^{\frac{1}{2}}=10\cdot10^{\frac{1}{2}}\cdot(10\sqrt[3]{...})^{{\frac{1}{2}}\cdot{\frac{1}{3}}}=10^{\frac{1}{2}}\cdot10^{{\frac{1}{2}}\cdot{\frac{1}{3}}}\cdots10^{\frac{1}{n!}}$$ Now, using the rule of exponents, we have: $$S=10^{\sum{i=1}^{\infty}\frac{1}{i!}}=10^{e-1}$$ because $$\lim_{\xi\to\infty}\sum_{i=1}^\xi\dfrac1{i!}=e-1$$
$$\lim_{n\to\infty}10^{\sum_{r=1}^n\dfrac1{r!}}=10^{\lim_{n\to\infty}\sum_{r=1}^n\dfrac1{r!}}$$
Now $\lim_{n\to\infty}\sum_{r=1}^n\dfrac1{r!}=e-1$