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## 'Harmonically' printed from http://nrich.maths.org/

(a) This is a divergent series, which means that the sum grows to
infinitiy. To show this, consider splitting the terms to be added
together into small chunks, each twice as long as the previous
one

$$

(1), (2,3), (4, 5, 6, 7), (8,9,10,11,12,13,14,15) \mbox{ etc}

$$

That is add the 'chunk' of the 2nd and 3rd terms, add
the'chunk' of the 4th, 5th, 6th and 7th, etc.
etc.

(b) You have to show that $$S_{n-1} > \log n > S_n - 1$$ so
that, for large $n$, $S_n$ is approximately equal to $\log n$.