Compute $\lim_{n\to\infty}(\frac{1+h^{\frac{1}{n}}}{2})^{n}$

Question

Compute $$L=\lim_{n\to\infty}\left(\frac{1+h^{\frac{1}{n}}}{2}\right)^{n}$$ where $h>0$.

With a computer to compute it numerically, $L$ seems to tend to $\sqrt{h}$, but I could not figure it out analytically.

The almighty Wolfram Mathematica confirms that result, so I guess it should be possible to derive it symbolically. TIA

Hint: Take logarithms and use $\log(1+x) \to x$ as $x \to 0$. — sudeep5221, Jul 07 '24 at 15:17

score 1 · Answer 1 · answered Jul 07 '24 at 15:32

1

One can rewrite the expression in the OP as $$A_n=\Big(1+\frac{n(1+h^{1/n}-2)}{2n}\Big)^n$$

$n(h^{1/n}-1)=\frac{h^{1/n}-1}{1/n}\xrightarrow{n\rightarrow\infty}\log h$

Hence $A_n\xrightarrow{n\rightarrow\infty}e^{\frac12\log h}=\sqrt{h}$

answered Jul 07 '24 at 15:32

Mittens

1 Answers1