Find limlimits_(n→∞)sqrt[n]1+i

Find $\ \lim\limits_{n\to\infty}\sqrt[n]{1+i}$.

Step 1 · Polar form

$$ 1+i=\sqrt2\,e^{i\pi/4} $$

Step 2 · The $n$-th root

$$ \sqrt[n]{1+i}=\left(\sqrt2\,e^{i\pi/4}\right)^{1/n}=2^{1/(2n)}\,e^{i\pi/(4n)} $$

Step 3 · Limits of modulus and argument

$$ \lim_{n\to\infty}2^{1/(2n)}=2^0=1, \qquad \lim_{n\to\infty}\frac\pi{4n}=0 $$

Step 4 · Combine

$$ \boxed{\lim_{n\to\infty}\sqrt[n]{1+i}=1\cdot e^{i\cdot0}=1} $$

General principle

For any fixed $z=re^{i\theta}\neq0$: $\ \sqrt[n]{z}=r^{1/n}e^{i\theta/n}\to1\cdot e^{0}=1$, since $r^{1/n}\to1$ and $\theta/n\to0$ regardless of the fixed values $r,\theta$.

Verification

$$ \sqrt[2]{1+i}=2^{1/4}e^{i\pi/8}\approx1.189\,e^{i\pi/8}, \qquad \sqrt[10]{1+i}=2^{1/20}e^{i\pi/40}\approx1.035+0.081i $$

both modulus and argument are visibly shrinking toward $1$ and $0$ as $n$ grows.