Don't forget about the constraints!

Algebra Level 4

Does there exist a real x x such that x + x 2 + x 3 + x 4 + x 5 + x 6 ⋯ x + x^2 + x^3 + x^4 + x^5 + x^6 \cdots is smaller than x − x 2 + x 3 − x 4 + x 5 − ⋯ x - x^2 + x^3 - x^4 + x^5 - \cdots ?

No Yes

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Chew-Seong Cheong
Nov 27, 2016

Assuming that it is true that there exists a real x x such that:

x + x 2 + x 3 + x 4 + x 5 + x 6 + . . . < x − x 2 + x 3 − x 4 + x 5 − x 6 + . . . ⟹ x 2 + x 4 + x 6 + x 8 . . . < − x 2 − x 4 − x 6 − x 8 − . . . 2 ( x 2 + x 4 + x 6 + x 8 . . . ) < 0 ⟹ x 2 + x 4 + x 6 + x 8 . . . < 0 \begin{aligned} x+x^2+x^3+x^4+x^5+x^6+... & < x-x^2+x^3-x^4+x^5-x^6+... \\ \implies x^2+x^4+x^6+x^8... & < -x^2-x^4-x^6-x^8-... \\ 2 ( x^2+x^4+x^6+x^8...) & < 0 \\ \implies x^2+x^4+x^6+x^8... & < 0 \end{aligned}

which is not true and it contradicts the assumption. N o \boxed{No} , therefore, is the answer.

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