5. Summations, Series and expansion of Functions.

When calculating a partition function, working out the dissociation energy of a molecule by counting energy levels, or calculating a Madelung constant, a series of terms must be summed. A different situation occurs when it is necessary to simplify an expression so as to be able to effect a solution, for example, if \(\displaystyle \sin(x)\) appears in a formula, for small \(x\) this can be approximated as \(x\) in other words the change \(\displaystyle\sin(x) \rightarrow x\). This can lead to considerable simplification such as in the motion of a pendulum. In other cases the assumption is made that it is possible to expand an expression as a Taylor or Maclaurin series and thereby learn something about regions in which we have no information. This assumption is based on the understanding that any function used varies in a slow and predictable way and that the extrapolation is not taken too far. In this chapter, both approaches are described.