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Omni-MATH / Let k be an integer greater than 1. Suppose a₀>0, and define a_(n+1)=a_(n)+(1)/(root[k](a_(n))) for n>0. Evaluate lim_(n→∞)(a_(n)^(k+1))/(n^(k)).

Problem

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problem

Let kk be an integer greater than 1. Suppose a0>0a_0 > 0, and define an+1=an+1anka_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}} for n>0n > 0. Evaluate limn→∞ank+1nk.\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.
Plain-text mathematical notation (without MathML)
Let k be an integer greater than 1. Suppose a₀>0, and define a_(n+1)=a_(n)+(1)/(root[k](a_(n))) for n>0. Evaluate lim_(n→∞)(a_(n)^(k+1))/(n^(k)).
Original LaTeX notation
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}}\] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\]

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Source and history

task_id
545075c6-5d52-58a2-b574-c1f1c36ca24f
task_key
test--545075c6-5d52-58a2-b574-c1f1c36ca24f
task_revision_id
1
config / split
/ test

Official source

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