Let $Y_n \sim Bin(n, 1/\sqrt n)$. Show that \begin{equation} \sqrt[4]{n}(Y_n/\sqrt{n} - 1) \xrightarrow{D} N(0,1) \end{equation} using that \begin{equation} X_n \xrightarrow{D} X \quad \text{ if and only if } \quad \forall t\in \mathbb R: \varphi_{X_n}(t) \to \varphi_X(t) \end{equation} for random variables $X_n, X$ having characteristic functions $\varphi_{X_n}(t)$, $\varphi_X(t)$, respectively.
I have tried the following: Take $t \in \mathbb R$ arbitrary. Using the properties of characteristic functions, we have \begin{align*} \varphi_{\sqrt[4]{n} (Y_n/\sqrt n - 1)}(t) &= \varphi_{Y_n/\sqrt n - 1}(\sqrt[4]{n} t) \\ &= \exp\{ -i \sqrt[4]{n} t \} \varphi_{Y_n/\sqrt n}(\sqrt[4]{n} t) \\ &= \exp\{ -i \sqrt[4]{n} t \} \varphi_{Y_n}(t / \sqrt[4]{n}) \\ &= \exp\{ -i \sqrt[4]{n} t \} \left( 1 - \frac{1}{\sqrt n} + \frac{1}{\sqrt n} e^{it / \sqrt[4]{n}} \right)^n. \end{align*}
The characteristic function of $Z \sim N(0,1)$ is $\varphi_Z(t) = \exp\{-\tfrac{1}{2}t^2\}$. We may equivalently show the convergence of the logarithms, i.e. that \begin{equation} -i \sqrt[4]{n} t + n \log\left( 1 - \frac{1}{\sqrt n} + \frac{1}{\sqrt n} e^{it / \sqrt[4]{n}} \right) \xrightarrow{n\to\infty} -\tfrac{1}{2} t^2. \end{equation}
This is really where I don't see what to do next. Any help would be much appreciated!