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2ntl Tπ b. (-1) 22n(2011): (20t1):

2ntl Tπ b. (-1) 22n(2011): (20t1):

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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Find the sum of the series
The image contains a mathematical expression written using summation notation, typically used in advanced mathematics or calculus. ### Expression: The expression is given as: \[ \text{(b)} \quad \sum_{n=0}^{8} \frac{(-1)^{n+1} \cdot \pi^{2n+1}}{2^{2n+1} \cdot (2n+1)!} \] ### Explanation: 1. **Summation Notation**: - The expression makes use of the sigma notation (∑), which indicates that you must add up all the terms generated by the expression for each integer value of \( n \) from the lower limit (in this case, 0) to the upper limit (in this case, 8). 2. **Components of the Expression**: - \( (-1)^{n+1} \): Alternating sign for each term. This component results in alternating series terms that switch every time \( n \) increases by 1. - \( \pi^{2n+1} \): Represents the powers of π raised to an odd integer power, starting from \(\pi^1\) when \( n = 0 \). - \( 2^{2n+1} \): This is the denominator part involving powers of 2, similar to the numerator, it raises 2 to an odd integer power. - \( (2n+1)! \): Factorial notation for odd integers, which is part of the denominator. Factorials are products of an integer and all the integers below it. ### Use: This type of expression can be part of solving infinite series, approximating functions, or calculating values like trigonometric series.
Transcribed Image Text:The image contains a mathematical expression written using summation notation, typically used in advanced mathematics or calculus. ### Expression: The expression is given as: \[ \text{(b)} \quad \sum_{n=0}^{8} \frac{(-1)^{n+1} \cdot \pi^{2n+1}}{2^{2n+1} \cdot (2n+1)!} \] ### Explanation: 1. **Summation Notation**: - The expression makes use of the sigma notation (∑), which indicates that you must add up all the terms generated by the expression for each integer value of \( n \) from the lower limit (in this case, 0) to the upper limit (in this case, 8). 2. **Components of the Expression**: - \( (-1)^{n+1} \): Alternating sign for each term. This component results in alternating series terms that switch every time \( n \) increases by 1. - \( \pi^{2n+1} \): Represents the powers of π raised to an odd integer power, starting from \(\pi^1\) when \( n = 0 \). - \( 2^{2n+1} \): This is the denominator part involving powers of 2, similar to the numerator, it raises 2 to an odd integer power. - \( (2n+1)! \): Factorial notation for odd integers, which is part of the denominator. Factorials are products of an integer and all the integers below it. ### Use: This type of expression can be part of solving infinite series, approximating functions, or calculating values like trigonometric series.
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