Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$
Prove with induction that, if $E(|X|^n)< \infty$, then $\varphi$ is $n$-times continuously differentiable and: $$ \varphi^{(k)}(0)=i^{k}\int_\mathbb{R} x^{k}d\mu(x),\;\;\;\; k=0,1,2,3,\dots,n $$
I have managed a proof for $k=1$, but im stuck with any values of $k\geq 2$. Please help
Hint: Interchange integration and differentiation, i.e. use that $$\frac{d^k}{d^k t} \int e^{\imath \, t x} \, d\mu(x) = \int \frac{d^k}{d^k t} e^{\imath \, t x} \mu(dx).$$ (Here $\frac{d^k}{d^k t}$ denotes the $k$-th derivative with respect to $t$.)
Remark: It follows from the theorem on differentiation of parametrized integrals and the integrability condition $\int |x|^n \, d\mu < \infty$ that we are indeed allowed to interchange integration and differentiation. The theorem states that if we consider an integral of the form
$$V(t) := \int g(t,x) \, d\mu(x)$$
where $g(t,\cdot) \in L^1(\mu)$ for all $t$ and $|\partial_t g(t,x)| \leq w(x)$ for some integrable function $w \in L^1(\mu)$, then
$$\frac{\partial}{\partial t} V(t) = \int \frac{\partial}{\partial t} g(t,x) \, d\mu(x).$$