Many sine functions

Adding many sine functions yields a so-called Fourier series (see page 1074). The pictures below show Sum[Sin[n x]/n, {n, k}] for various numbers of terms k. Apart from a glitch that gets narrower with increasing k (the so-called Gibbs phenomenon), the result has a simple triangular form. Other so-called Fourier series in which the coefficient of Sin[m x] is a smooth function of m for all integer m yield similarly simple results.

The pictures below show Sum[Sin[n^{2} x]/n^{2}, {n, k}], where in effect all coefficients of Sin[m x] other than those where m is a perfect square are set to zero. The result is a much more complicated curve. Note that for x of the form p π/q, the k=∞ sum is just

(π/(2q))^{2} Sum[Sin[n^{2} p π/q]/Sin[n π/(2q)]^{2},{n,q-1}]

The pictures below show Sum[Cos[2^{n} x], {n, k}] (as studied by Karl Weierstrass in 1872). The curves obtained in this case show a definite nested structure, in which the value at a point x is essentially determined directly from the base 2 digit sequence of x. (See also page 1080.)

The curves below are approximations to Sum[Cos[2^{n} x]/2^(a n), {n, ∞}]. They can be thought of as having dimensions 2-a and smoothed power spectra ω^-(1+2a).