Techniques are developed to find all periodic solutions in the simple pendulum by means of the homotopy analysis method (HAM). This involves the solution of the equations of motion in two different coordinate representations. Expressions are obtained for the cycles and periods of oscillations with a high degree of accuracy in the whole range of amplitudes. Moreover, the convergence of the method is easily checked. The aim of this work is to show how the dynamics of a simple example of oscillatory systems may be studied globally with the HAM and to incentivize the interest of advanced undergraduate students in this type of techniques. (Contains 3 figures.)