Denote-x-n-is-the-unique-positive-root-of-the-following-equation-x-n-x-n-1-x-n-2-Prove-that-the-sequence-x-n-converges-to-a-positive-real-number-Find-that-limit-

Question Number 153897 by mathdanisur last updated on 11/Sep/21

Denote x_{n} is the unique positive root

of the following equation :

x^{n} + x^{n - 1} + \dots x = n + 2

Prove that the sequence (x_{n}) converges

to a positive real number . Find that

limit .

Commented by MJS_new last updated on 12/Sep/21

this is not a proof but \dots

f_{n} (x) = x^{n} + x^{n - 1} + \dots + x

\Rightarrow

f_{n} (1) = n

\Rightarrow

f_{n} (1 + k) > n \forall k > 0

\Rightarrow

\exists k > 0 : f_{n} (1 + k) = n + 2

obviously

n \to \infty \Rightarrow k \to 0

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