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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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How Many Solutions?

Find all the solutions to the this equation.

Climbing Powers

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

We can define $2^{3^{4}}$ either as $(2^{3})^{4}$ or as $2^{(3^{4})}$ . Does it make any difference?

Now calculate $\left(\sqrt 2^{ \sqrt 2 }\right)^{ \sqrt 2 }$ and $\sqrt 2 ^{\left(\sqrt 2 ^{ \sqrt 2 }\right)}$ and answer the following question for the natural extension of both definitions.

Which number is the biggest \[ \sqrt 2 ^{\sqrt 2 ^{\sqrt 2 ^{\sqrt 2 ^{.^{.^{.}}}}}} \]

where the powers of root $2$ go on for ever, or \[ \left(\sqrt 2 ^{\sqrt 2 }\right)^{\sqrt 2} ? \]