The aims of this option are to introduce limit theorems and convergence of series, and to use calculus results to solve differential equations. Before beginning any work in this option, it is recommended that you revise Topic 1 and Topic 7 of the core syllabus, as a lot of background knowledge of those topics are helpful in this topic.

The Harmonic Series

A harmonic series is a divergent infinite series. An example of which is:

${\displaystyle S_{n}=\sum _{n=1}^{\infty }{\frac {1}{n}}}$

where, the following pattern is observed:

• ${\displaystyle s_{1}=1}$

• ${\displaystyle {s_{2}=1+{\frac {1}{2}}}={\frac {3}{2}}}$

• ${\displaystyle {s_{4}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}}>1+{\frac {1}{2}}+{\frac {1}{2}}=2}$

• ${\displaystyle {s_{8}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}}>1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}=2{\frac {1}{2}}}$

• ${\displaystyle {s_{16}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}+{\frac {1}{10}}+{\frac {1}{11}}+{\frac {1}{12}}+{\frac {1}{13}}+{\frac {1}{14}}+{\frac {1}{15}}+{\frac {1}{16}}}>1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}=3}$

From this one can conclude that the pattern will countinue:

${\displaystyle s_{32}>3{\frac {1}{2}}}$ and ${\displaystyle s_{64}>4}$,

Thus the general pattern can be expressed as:

${\displaystyle s_{2^{n}}>1+{\frac {n}{2}}}$