IB Mathematics (HL)/Series and Differential Equations

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:

$S_n=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$

where, the following pattern is observed:

• $s_1=1$

• $s_2=1+\frac{1}{2}=\frac{3}{2}$

• $s_4=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}>1+\frac{1}{2}+\frac{1}{2}=2$

• $s_8=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}>1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2\frac{1}{2}$

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

From this one can conclude that the pattern will countinue:

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

Thus the general pattern can be expressed as:

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