A calculus of the absurd

3.4.2 Sum to infinity

The sum to infinity (which I think is in the formula booklet) doesn’t always converge to a specific value (because the terms get bigger and bigger). For example, the value of

\begin{equation} 1 + 2 + 4 + 8 + ... \end{equation}

is infinite, but the value of

\begin{equation} 1 + \frac {1}{2} + \frac {1}{4} + \frac {1}{8} + ... \end{equation}

is not infinite, and can be found. If the size (i.e. it doesn’t matter if the value is positive or negative) of the common ratio is less than one, then the terms in the series become successively smaller and smaller (note that just because the size of the terms in a series decreases does NOT mean that it converges to a fixed value, for example the series \(1 + \frac {1}{2} + ... + \frac {1}{n} + ...\) does not converge). This is because multiplying two decimal numbers together makes a smaller number ⁴⁴ e.g. \(\frac {1}{4} \cdot \frac {1}{4} = \frac {1}{16}\) and \(\frac {1}{16}\) is smaller than \(\frac {1}{4}\).

If \(\abs {r} < 1\) (i.e. if \(r\) is greater than \(-1\) and less than \(1\)), then as \(n\) tends to infinity (written \(n \to \infty \)) then \(r^n\) tends to \(0\) (written \(r^n \to 0\)). We can apply this to Equation 3.49 and obtain that

\begin{equation} S_{\infty } = \frac {a}{1-r} \end{equation}