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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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Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

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Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Giants

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2
Note that you don't need to work out the numbers exactly to decide which is bigger!

Some possible methods:

1) Can you split up the powers into smaller pieces, or consider another quantity which your calculator can handle from which you can deduce the answer?

or 2) Can you use inequalities and logic such as 'if X > Y and Y > Z then we know that X > Z'

or 3) What happens when you raise $100^{1/100}$ to the power $9900$?