What is the series representation of $\int_{0}^{t} \exp(-x^n)\, dx$ where $n$ is a positive integer? I have searched in the web to get series representation of $$\int_{0}^{t} e^{-x^n}\,dx$$  where $n$ is a positive integer, but I haven’t succeeded; really I want its representation as hypergeometric series.

I am unfamiliar with hypergeometric series (such as this?), but since in the comments you indicated that a Riemann sum could suffice, here is a brute-force approach:

Consider that by definition $$\int_a^b f(X) \, dX = \lim_{N\to\infty} \sum_{i=1}^N f(x_i) \, \Delta x$$ where $x_i$ could be the righthand points of $N$ subintervals $[a,b]$ each of width $\Delta x$.
For your example, $a=0$ and $b=t$. Therefore the length of the domain of integration is merely $b-a=t$, and when this is partitioned into $N$ subintervals, $\Delta x = t/N$.
The righthand $x_i$ of each Riemannian rectangle is given by $a+i\,\Delta x$, in this case $x_i = it/N$.
Hence $$\int_0^t \exp\left( -x^n\right) \, dx = \lim_{N\to\infty}\sum_{i=1}^N \exp\left[ -\left(\frac{it}{N}\right)^n\right] \frac tN$$

If you are interested in pursuing this approach further I can try to continue the evaluation and/or construct a visual if necessary.
