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Omni-MATH / Consider the following sequence…

Problem

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problem

Consider the following sequence (an)n=1∞=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,…)\left(a_{n}\right)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \ldots) Find all pairs (α,β)(\alpha, \beta) of positive real numbers such that limn→∞∑k=1naknα=β\lim _{n \rightarrow \infty} \frac{\sum_{k=1}^{n} a_{k}}{n^{\alpha}}=\beta.
Plain-text mathematical notation (without MathML)
Consider the following sequence (a_(n))_(n=1)^(∞)=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,…) Find all pairs (α,β) of positive real numbers such that lim_(n→∞)(∑_(k=1)^(n)a_(k))/(n^(α))=β.
Original LaTeX notation
Consider the following sequence $$\left(a_{n}\right)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \ldots)$$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim _{n \rightarrow \infty} \frac{\sum_{k=1}^{n} a_{k}}{n^{\alpha}}=\beta$.

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Source and history

task_id
2e2754a2-c514-5a22-b00f-c1862206eca4
task_key
test--2e2754a2-c514-5a22-b00f-c1862206eca4
task_revision_id
1
config / split
/ test

Official source

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