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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 $S_1, S_2, ... S_{r-1}$.

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Degree Ceremony

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

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OK! Now Prove It

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

Harmonically

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

(a) Is it true that a large value of $n$ can be found such that: $$S_n = 1 +{1\over 2} + {1\over 3} + {1\over 4} + ... + {1\over n} > 100?$$

(b) By considering the area under the graph of $y = {1\over x}$ between $a ={1\over n}$ and $b = {1\over n-1}$ show that this series grows like $\log n$.