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Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?


Age 16 to 18
Challenge Level

(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 $\log n$.