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I am trying to find the limit of the sequence $(a_n)_{n\geq 1}$ where $a_n = \sqrt[n]{ 5^n+11^n+17^n}$. I am really stuck on this as I should not be allows to use exponential and logarithmic properties to solve this. Through graphing I have found that $\lim_{n \to \infty} a_n = 17$, which can intuitively be understood as the $17^n$ absorbing the other terms, but I have not idea how to prove it.

I though of using Monotone convergence theorem but I am not able to show that $a_{n} \geq a_{n+1}$. Could I get a hint ?

BrockenDuck
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    Hint: $$ a_n = 17\sqrt[n]{{1 + \left( {\frac{5}{{17}}} \right)^n + \left( {\frac{{11}}{{17}}} \right)^n }}. $$ The expression under the root is always between $1$ and $2$. – Gary Mar 06 '22 at 12:02
  • More of this kind: https://math.stackexchange.com/q/2533102/42969 or https://math.stackexchange.com/q/2128024/42969 – Martin R Mar 06 '22 at 12:08
  • Gary's hint is the standard approach for such exercises. Usually, factoring out the term with the fastest growth rate solves the problem. – Peter Mar 06 '22 at 12:10

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