Suppose like the infinite series of the exponential function e^x.
its maclaurin series expansion is x^n/n! where the terms can be written out as x + x^2/2 + x^3/6 and so on.
so taking the derivative of it would then be 1 + x + x^2/2...+ x^n/n!
the pattern is the same as the original function, and thus the infinite series of e^x is still e^x. I find this method easier than applying the formula where you have to find the sequence and memorize everything else.