 The rearranged series 1 − 1 − 1 + 1 + 1 − 1 − 1 + · · · occurs in Euler's 1775 treatment of the pentagonal number theorem as the value of the Euler function at "q" = 1.The power series most famously associated with Grandi's series is its ordinary generating function,

In his 1822 "Théorie Analytique de la Chaleur", Joseph Fourier obtains what is currently called a Fourier sine series for a scaled version of the hyperbolic sine function,

He finds that the general coefficient of sin "nx" in the series is

For "n" > 1 the above series converges, while the coefficient of sin "x" appears as 1 − 1 + 1 − 1 + · · · and so is expected to be 1⁄2.In fact, this is correct, as can be demonstrated by directly calculating the Fourier coefficient from an integral