6. Find A and B such that for all i > 2 1 A i2 + 5i + 6 i+2 i +3' Mol and use this identity to express the partial sum s, = 1/(2² + 5i +6) of the series 1 22+5i + 6 n21 as a telescoping sum. Use this telescoping sum to show first that the series is convergent, and find its sum.

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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**Problem Statement:**

6. Find \( A \) and \( B \) such that for all \( i \geq 2 \)

\[
\frac{1}{i^2 + 5i + 6} = \frac{A}{i+2} + \frac{B}{i+3}
\]

and use this identity to express the partial sum 

\[
s_n = \sum_{i=1}^{n} \frac{1}{i^2 + 5i + 6}
\]

of the series 

\[
\sum_{n \geq 1} \frac{1}{i^2 + 5i + 6}
\]

as a telescoping sum. Use this telescoping sum to show first that the series is convergent, and find its sum.
Transcribed Image Text:**Problem Statement:** 6. Find \( A \) and \( B \) such that for all \( i \geq 2 \) \[ \frac{1}{i^2 + 5i + 6} = \frac{A}{i+2} + \frac{B}{i+3} \] and use this identity to express the partial sum \[ s_n = \sum_{i=1}^{n} \frac{1}{i^2 + 5i + 6} \] of the series \[ \sum_{n \geq 1} \frac{1}{i^2 + 5i + 6} \] as a telescoping sum. Use this telescoping sum to show first that the series is convergent, and find its sum.
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