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Let $a,b$ be two numbers such that $0<a<b$. Consider the sequence $x_n=a^n+b^n$. Calculate $\displaystyle\lim_{n\to\infty}\frac{x_{n+1}}{x_n}$

I did the following algebraic manipulations: $\displaystyle\lim_{n\to\infty}\frac{x_{n+1}}{x_n}=\displaystyle\lim_{n\to\infty}\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ $$=\displaystyle\lim_{n\to\infty}\left(\frac{{a^{n+1}/b^n}+b}{a^n/b^n+1}\right)*\frac{b^n}{b^n}$$

However, I got stuck here.

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    $\begingroup$ You are on the right track. The limit is $b$. $\endgroup$
    – Idonknow
    Commented Oct 3, 2020 at 16:07
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    $\begingroup$ Try with this version$$\left(\frac{\left(\frac{a}{b}\right)^{n+1}+1}{\left(\frac{a}{b}\right)^n+1}\right)\cdot b$$ $\endgroup$
    – rtybase
    Commented Oct 3, 2020 at 16:09

1 Answer 1

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Hint : as , $$0\lt \frac{a}{b}\lt 1 $$ So, $$\lim_{n\to \infty} \left(\frac{a}{b}\right)^{n} = 0 $$

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