This question represents some thoughts about the following question: How to get PDF from characteristic function
In that question $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-itx}\phi(t)dt,$$ where $\phi(t)$ is a characteristic function.
What if it is hard to find this integral? But the Taylor series expansion of $\phi(t)$ is $$\phi(t)=\sum_{k=0}^{\infty}c_k(it)^k$$
Then $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-itx}\phi(t)dt=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-itx}\left(\sum_{k=0}^{\infty}c_k(it)^k\right)dt=\frac{1}{2\pi}\sum_{k=0}^{\infty}c_k\int_{-\infty}^{\infty}e^{-itx}(it)^k dt=\sum_{k=0}^{\infty}(-1)^k c_k \operatorname{dirac}^{(k)}(x)$$
Does it make any sense?
I think that I'm wrong somewhere because the sum of the derivatives of the Dirac delta function still gives zero on a plot.
Thank you.