Using the bevel tool is a simple way to do this. First remove all of the edge loops on the endcap so you just have a single n-gon like this:
Note that I scaled the cylinder to give it an elliptical profile and skewed the face so that it is not perpendicular to the axis of the cylinder. I did this because this is how it appears to me in your picture. If you make your endcap perpendicular to the profile axis you will get a cleaner (more spherical) result.
Be that as it may, I used the Bevel tool which is located in the menu: Mesh > Edges > Bevel with the settings shown on the bottom left of the following image:
Using 8 segments achieves a well rounded result, although as I previously mentioned, the dome shape is distorted because of the angle of the face in respect to axis of the profile. I found that the "Depth" setting worked the best in this situation because I was able to extend the bevel very close to the convergence point without getting overlap. You may want to experiment with the settings to get what you want.
Just to clarify what I mean about getting a cleaner, more spherical result by making the end cap perpendicular to the profile axis, here is a comparison: