The solution of the mathematical  problem

The answer is sure enough "NO", because it's sufficient to consider the numerical suite (S_n), of general  term :

S_n = 1/(n+1)   +   (-1)ⁿ/(n+2)  

In fact, for any positif integer n, it's clear that 1/(n+1) is strictly superior at  1/(n+2), therefore in the case where 1/(n+2) is preceded by a negatif sign ( in the other case, we haven't any problem !!) ; the sum is always postive, and consequently all the terms of the suite ( S_n ) are positive, and we can verify this :

^S₁=1/2  +  (-1)¹/3 = 1/2  -1/3 = 1/6 = 0.166^... >0;

^S₂=1/3  + (-1)²/4 = 1/3 + 1/4= 7/12  =0.583... >0;

^S₃=1/4 + (-1)³/5 = 1/4 -1/5 = 1/20 = 0.05 >0........etc.

In another side, the (lim S_n =0) when (n--> + ∞)"you can verify"

Therefore, all terms of this numerical suite are positive and ( lim S_n =0 ) when (n--> + ∞), but, we well see, for example that S₁ < S₂ and S₂ > S₃,

Therefore, the suite isn't monotonous, and consequently this suite prove that the hypothesis is wrong.

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