every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma. As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere. A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable ; see Laurent series. For example, can be written as a Laurent series. Generalization There is, however, a generalization of the Taylor series that does converge to the value of the function itself for any bounded continuous function on , using the calculus of finite differences. Specifically, one has the following theorem, due to Einar Hille, that for any , Here is the th finite difference operator with step size . The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function is analytic at , the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series. In general, for any infinite sequence , the following power series identity holds: So in particular, The series on the right is the expectation value of , where is a Poisson-distributed random variable that takes the value with probability . Hence, The law of large numbers implies that the identity holds. List of Maclaurin series of some common functions Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments . Exponential function The exponential function (with base ) has Maclaurin series . It converges for all . The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function: Natural logarithm The natural logarithm (with base ) has Maclaurin series They converge for . (In addition, the series for converges for , and the series for converges for .) Geometric series The geometric series and its derivatives have Maclaurin series All are convergent for . These are special cases of the binomial series given in the next section. Binomial series The binomial series is the power