For n ∈ N , let A n , G n and H n be the arithmetic, geometric and harmonic means of n + 1 , n + 2 , … , 2 n , respectively. Find lim n → ∞ H n / n .

n / H n = ∑ k = 1 n 1/ ( n + k ) → ∫ 0 1 dx / ( 1 + x ) = ln 2 . Hence H n / n → 1/ ln 2 = lo g 2 e.

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MathematicsSequence & SeriesMeans / inequalityMedium2 minQB

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For $n \in N$ , let $A_{n}, G_{n}$ and $H_{n}$ be the arithmetic, geometric and harmonic means of $n + 1, n + 2, \dots, 2 n$ , respectively. Find $lim_{n \to \infty} H_{n} / n$ .

Options:

Answer:

Solution:

$n / H_{n} = \sum_{k = 1}^{n} 1/ (n + k) \to \int_{0}^{1} dx / (1 + x) = ln 2$ . Hence $H_{n} / n \to 1/ ln 2 = lo g_{2}$ e.

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Means / inequality

⏱ 2mℹ️ Source: QB

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