Math Problem Statement

Let (a_n)_{n ≥ 1} be the sequence defined by a_n = 1/(n+1) + 1/(n+2) + ... + 1/(2n). Prove that (a_n) is increasing and bounded. Deduce the limit of the sequence.

Solution

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Math Problem Analysis

Mathematical Concepts

Sequences and Series
Limits
Monotonicity
Bounded Sequences

Formulas

a_n = Σ (1 / (k + 1)) for k = n to 2n - 1
ln(n+1) - ln(n) = ln(1 + 1/n)
Integral approximation

Theorems

Monotone Convergence Theorem
Properties of logarithmic functions

Suitable Grade Level

Undergraduate (Mathematics Major)

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