Progressed to infinity

Algebra Level 3

S = ∑ k = 2 ∞ [ 1 k + ( 1 k ) 2 + ( 1 k ) 3 + ⋯ ] = [ 1 2 + ( 1 2 ) 2 + ( 1 2 ) 3 + ⋯ ] + [ 1 3 + ( 1 3 ) 2 + ( 1 3 ) 3 + ⋯ ] + [ 1 4 + ( 1 4 ) 2 + ( 1 4 ) 3 + ⋯ ] + ⋯ \begin{aligned} S & = \sum_{k=2}^\infty \left[ \dfrac1k + \left( \dfrac1k\right)^2 + \left( \dfrac1k\right)^3 + \cdots \right] \\ & = \left[ \dfrac12 + \left( \dfrac12\right)^2 + \left( \dfrac12\right)^3 + \cdots \right] + \left[ \dfrac13 + \left( \dfrac13\right)^2 + \left( \dfrac13\right)^3 + \cdots \right] \\ & \quad \quad + \left[ \dfrac14 + \left( \dfrac14\right)^2 + \left( \dfrac14\right)^3 + \cdots \right] + \cdots \end{aligned}

Compute the sum S S above.

It diverges to infinity 2 e 4 3

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Aug 27, 2016

By GP sum rule, the first term is 1, the second is 1/2, the third is 1/3, etc.

So we have the harmonic series: 1 + 1/2 + 1/3 + 1/4 + 1/5 +....

This sum is infinite. (Proof: it is greater than 1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8)... which is equal to 1 + 1/2 + 1/2 + 1/2 +.... which is infinite.)

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