En=1 2n –1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Determine if the following series converges or diverges.
![The image shows a mathematical series represented as:
\[
\sum_{n=1}^{\infty} \frac{1}{2n - 1}
\]
This denotes the sum of the reciprocals of all odd numbers starting from 1. The series is an infinite series where each term in the series is obtained by the formula \(\frac{1}{2n - 1}\), with \(n\) taking integer values starting from 1 and continuing indefinitely.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a3bdd75-f939-43d0-9bcd-1717dd46199e%2Fc16a1eda-280d-40a2-88f2-22f94603e981%2Fxrljvln_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical series represented as:
\[
\sum_{n=1}^{\infty} \frac{1}{2n - 1}
\]
This denotes the sum of the reciprocals of all odd numbers starting from 1. The series is an infinite series where each term in the series is obtained by the formula \(\frac{1}{2n - 1}\), with \(n\) taking integer values starting from 1 and continuing indefinitely.
![The image shows a mathematical series expressed as:
\[
\sum_{n=1}^{\infty} \frac{1 + \cos(n)}{e^n}
\]
This represents an infinite sum where each term is given by \((1 + \cos(n))/e^n\), and the summation index \(n\) starts at 1 and goes to infinity.
In this series:
- \(\cos(n)\) is the cosine of \(n\), a trigonometric function that oscillates between -1 and 1.
- \(e^n\) is the exponential function where \(e\) is the base of natural logarithms, approximately equal to 2.71828.
- The numerator \(1 + \cos(n)\) modifies the amplitude of the function.
- The denominator \(e^n\) indicates that each term in the series decreases exponentially as \(n\) increases, due to the rapidly growing nature of the exponential function.
This summation is typically studied in the context of convergence of series, analyzing whether the series sums to a finite number as \(n\) approaches infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a3bdd75-f939-43d0-9bcd-1717dd46199e%2Fc16a1eda-280d-40a2-88f2-22f94603e981%2F4sdkmu8_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical series expressed as:
\[
\sum_{n=1}^{\infty} \frac{1 + \cos(n)}{e^n}
\]
This represents an infinite sum where each term is given by \((1 + \cos(n))/e^n\), and the summation index \(n\) starts at 1 and goes to infinity.
In this series:
- \(\cos(n)\) is the cosine of \(n\), a trigonometric function that oscillates between -1 and 1.
- \(e^n\) is the exponential function where \(e\) is the base of natural logarithms, approximately equal to 2.71828.
- The numerator \(1 + \cos(n)\) modifies the amplitude of the function.
- The denominator \(e^n\) indicates that each term in the series decreases exponentially as \(n\) increases, due to the rapidly growing nature of the exponential function.
This summation is typically studied in the context of convergence of series, analyzing whether the series sums to a finite number as \(n\) approaches infinity.
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