As useful as sequences and series are, the real "power" in series comes when we introduce a variable  x  into the series.  Up to this point in time, the only variable in our series has been the index  n  running from 1 to infinity.  In power series, we keep the index  n  as before and introduce powers of  x,  i.e., xⁿ, into the nth term of the series, hence the name power series.

Many popular functions like trigonometric functions and rational functions have power series representations.  A polynomial and its power series representation are one and the same.  Some functions like the Bessel functions are defined in terms of its power series representations.

The famous Maclaurin series and Taylor series, special cases of power series, have wide applications in physics and engineering.

• Determine the radius and interval of convergence of a given power series.
• Use operations on a given power series to create a new power series.
• Use Taylor's theorem to create a power series for a given function.
• Find Taylor polynomials of a given function and use them to approximate the values of the function.