For most of the time that I have had a class using partial fractions, I've struggled with the normal method. I'm not sure if the method I've linked below is necessarilly better or new, but it's the one that I instinctively learned. There's way you can extend it using polynomial divison, but I do not have a clear way of explaining it.
Partial Fractionsp.s. I'll also admit that this was for a college project and may not be the best of instructions.
This is something I stumbled on while trying to make discreet sums into continuous functions. While it was already discovered by Muller, I'm still proud to have derived it myself, as simply typing in "Muller's Formula" into google gives about 30 different things named after him, none of which reference this formula beyond Wikipedia.
The second sum effectively reveals consecutive terms of the series as x increases. For non integer values for x, this effectively leads to interpolation between partial sums. The first term effectively acts as an error correction for this, as the formula isn't truly infinite. A derrivation for these formula can be found here.
Honestly, I figured this out purely as a way to make headaches. It's correct as far as I can tell (Actually, while writing this, I noticed that there is a missing n factorial out front.) but completely impractical. The one way I could see this being useful is as a way to explain how the derivative can carry information about the function that doesn't involve the concept of instantanieous change, which seems to trip people up when they're trying to understand it. Is this actually helpful in achieving that? Probably not :P