The Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivates at a single point. Its utility could be to graphically represent functions by selecting a sufficient number of terms and thus decreasing the error gradually.
or, in expanded form:
For the case , the series is a particular case of the Taylor Series, called the Maclaurin series.
Find the Maclaurin expansion for :
which is a repeating pattern.
Now we expand the coefficients:
observing that certain nominators contain and can be reduced, we obtain the final formula for :
Using the alternating series procedure, the series converges because: