The Binomial Series
The Binomial Theorem If k is any real number and |x|<1, then
The Binomial Series If k is any real number and |x|<1, then
where
The radius of convergence is 1, but the interval of convergence depends on the value of k.
The binomial series expansion is another of Newton's triumphs when he was about 22 years old (he also invented calculus, described his theory of colors, and described his laws of motion and the law of universal gravitation, while sitting out a plague afflicting London - under an apple tree, of course!). The Chinese knew the formula for the coefficients in the binomial theorem for positive integers long before, but Newton extended it to all real exponents. Pascal's triangle is a simple recursive exercise to get the coefficients in the binomial theorem for the case of positive integer exponents.
This is a pretty straightforward example of a Taylor's series expansion for a whole family of important functions.