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Find the constants of a limit

Source: CIIM 2023 - Problem 6

September 19, 2023

limitalgebrareal analysisinequalities

Problem Statement

Let

n

be a positive integer. We define

f(n)

as the number of finite sequences

(a_1, a_2, \ldots , a_k)

of positive integers such that

a_1 < a_2 < a_3 < \cdots < a_k

and

a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.

Determine the positive constants

\alpha

and

C

such that

\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.

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