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77. an = (In 3 In 2) + (ln 4 − ln 3) + (ln 5 − ln 4) + - - + (In (n − 1) - ln (n − 2)) + (lnn − ln(n − 1)) - - -

77. an = (In 3 In 2) + (ln 4 − ln 3) + (ln 5 − ln 4) + - - + (In (n − 1) - ln (n − 2)) + (lnn − ln(n − 1)) - - -

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question

Find if the sequence converges or diverges and state the limit of a convergent sequence. The given sequence is below: 

**Problem 77:** Evaluate the series given by \( a_n \): \[ a_n = (\ln 3 - \ln 2) + (\ln 4 - \ln 3) + (\ln 5 - \ln 4) + \cdots + (\ln (n - 1) - \ln (n - 2)) + (\ln n - \ln (n - 1)) \] Breaking down the expression for \(a_n\) into its components, you notice that each term of the form \( (\ln k - \ln (k-1)) \) for \( k = 3 \) to \( k = n \). This implies that each negative logarithm term cancels with the positive logarithm term of the subsequent term in the series. Consequently, this series is telescoping. By cancellation, it simplifies to: \[ a_n = \ln n - \ln 2 = \ln \left( \frac{n}{2} \right) \] Hence, the final value for \( a_n \) is: \[ a_n = \ln \left( \frac{n}{2} \right) \]
Transcribed Image Text:**Problem 77:** Evaluate the series given by \( a_n \): \[ a_n = (\ln 3 - \ln 2) + (\ln 4 - \ln 3) + (\ln 5 - \ln 4) + \cdots + (\ln (n - 1) - \ln (n - 2)) + (\ln n - \ln (n - 1)) \] Breaking down the expression for \(a_n\) into its components, you notice that each term of the form \( (\ln k - \ln (k-1)) \) for \( k = 3 \) to \( k = n \). This implies that each negative logarithm term cancels with the positive logarithm term of the subsequent term in the series. Consequently, this series is telescoping. By cancellation, it simplifies to: \[ a_n = \ln n - \ln 2 = \ln \left( \frac{n}{2} \right) \] Hence, the final value for \( a_n \) is: \[ a_n = \ln \left( \frac{n}{2} \right) \]
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