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[n2 x]/n2, {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/q/(2q))2 Sum[Sin[n2 p π/q]/Sin[n π/(2q)]2, {n, q - 1}]
The pictures below show Sum[Cos[2n 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[2n x]/2a n, {n, ∞}]. They can be thought of as having dimensions 2 - a and smoothed power spectra ω-(1 + 2a).