Asymptotic expansion of $n(\sqrt[n]{a} - 1)$, if $a > 0$, to terms of $O\Big(\dfrac{1}{n^3}\Big)$
Attempt:
$n((1+(a-1))^\frac{1}{n} - 1) = n(1 + \frac{1}{n} a + \dfrac{\frac{1}{n}(\frac{1}{n}-1)}{2} a^2+ \cdots - 1)$
$n(\frac{1}{n} a + \dfrac{\frac{1}{n}(\frac{1}{n}-1)}{2} a^2 + \cdots ) = a + \frac{\frac{1}{n}-1}{2} a^2 + \cdots$
Hint: $$ \sqrt[n]{a}=e^{\frac1n\log a}=1+\frac1n\,\log a+\frac1{2!}\,\frac1{n^2}(\log a)^2+\dots $$