The binomial theorem for positive integer exponents can be generalized to negative integer exponents. This gives rise to several familiar Maclaurin series with numerous applications in calculus and other areas of mathematics.
is not a polynomial. While positive powers of can be expanded into polynomials, e.g. , cannot be, so there cannot be a finite sum of monomial terms that equals . But there is a way to recover the same type of expansion if infinite sums are allowed.
As a first approximation, since by the power rule, the tangent line at is . So for small , This approximation is already quite useful, but it is possible to approximate the function more carefully using series.
Expand as a Maclaurin series.
The Maclaurin series for , wherever it converges, can be expressed as
Let . Applying the power rule repeatedly, we have
So the Maclaurin series becomes
This converges for by the ratio test.
The above example generalizes immediately for all negative integer exponents . Let be a real number and a positive integer. Define
then the same analysis as in the example gives
Let be a positive integer. Then for .