MathDB

For each real number

x

, let

\lfloor x \rfloor

denote the largest integer not exceeding

x

. A sequence

\{a_n \}_{n=1}^{\infty}

is defined by

a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.

Let

b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.

a) Find a polynomial

P(x)

with real coefficients such that

b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1

. b) Prove that there exists a strictly increasing sequence

\{n_k \}_{k=1}^{\infty}

of positive integers such that

\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.

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