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## Tuesday, March 23, 2010

### Proof that harmonic series diverges

Let $H_n=\sum_{k=1}^n{\frac{1}{k}}$
We investigate $H_{2^k}$
$H_{2^0}=H_1=\sum_{k=1}^1{\frac{1}{k}}=\frac{1}{1}=1+0(\frac{1}{2})$
$H_{2^1}=H_2=\sum_{k=1}^2{\frac{1}{k}}=\frac{1}{1}+\frac{1}{2}=1+1(\frac{1}{2})$
\begin{align*} H_{2^2} &=H_4\\ &=\sum_{k=1}^4{\frac{1}{k}}\\ &=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\\ &\gt \frac{1}{1}+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})\\ &\gt \frac{1}{1}+\frac{1}{2}+\frac{1}{2}\\ &\gt1+2(\frac{1}{2}) \end{align*}

\begin{align*} H_{2^3} &=H_8\\ &=\sum_{k=1}^8{\frac{1}{k}}\\ &=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\\ &\gt\frac{1}{1}+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8})\\ &\gt \frac{1}{1}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\\ &\gt1+3(\frac{1}{2}) \end{align*}
By induction it can be shown that
$H_{2^k}\gt1+k(\frac{1}{2})$
Since we can make $k(\frac{1}{2})$ as large as we want by choosing a value for $k$ this implies that we can also make $H_{2^k}$ as large as want, i.e. $H$ diverges.

Started on MT365-TMA01/CMA41. Started on question 1 and got lost in the mechanics of GraphPlot and it's options. Learned some Mathematica again. I have to increase my efforts on MT365. Yellow Alert.

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Mathematics: is it the fabric of MEST?
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)