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8 1 ₁ }(s) = = nº n=1 Show that finding the value of the Reiman Zeta for (2) is just the value Basel Problem, and that for all s> 1, that (s) exists.

8 1 ₁ }(s) = = nº n=1 Show that finding the value of the Reiman Zeta for (2) is just the value Basel Problem, and that for all s> 1, that (s) exists.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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### Understanding the Riemann Zeta Function The Riemann zeta function, denoted as ζ(s), is defined by the following infinite series: \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \] where \( s \) is a complex number with a real part greater than 1. ### Exploration Task 1. **The Basel Problem Connection:** - Evaluate the Riemann zeta function at \( s = 2 \), i.e., find \( \zeta(2) \). This evaluation is known to align with the solution to the Basel Problem. 2. **Existence for \( s > 1 \):** - Demonstrate that the Riemann zeta function \( \zeta(s) \) exists and is well-defined for all \( s \) with a real part greater than 1. #### Key Points - **Basel Problem:** Historically, the Basel Problem refers to the challenge of finding the exact sum of the reciprocals of the squares of the natural numbers. Mathematically, this is expressed as: \[ \sum_{n=1}^{\infty} \frac{1}{n^2} \] The solution to the Basel Problem is known, and evaluating \( \zeta(2) \) provides the exact value. - **Convergence of Series:** - For \( s \) with a real part greater than 1, the series converges, meaning the sum approaches a specific finite value. This property ensures that \( \zeta(s) \) is defined and exists for these values of \( s \). --- This content should assist learners in understanding the practical application of the Riemann zeta function in relation to the Basel Problem and the conditions under which the series converges.
Transcribed Image Text:### Understanding the Riemann Zeta Function The Riemann zeta function, denoted as ζ(s), is defined by the following infinite series: \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \] where \( s \) is a complex number with a real part greater than 1. ### Exploration Task 1. **The Basel Problem Connection:** - Evaluate the Riemann zeta function at \( s = 2 \), i.e., find \( \zeta(2) \). This evaluation is known to align with the solution to the Basel Problem. 2. **Existence for \( s > 1 \):** - Demonstrate that the Riemann zeta function \( \zeta(s) \) exists and is well-defined for all \( s \) with a real part greater than 1. #### Key Points - **Basel Problem:** Historically, the Basel Problem refers to the challenge of finding the exact sum of the reciprocals of the squares of the natural numbers. Mathematically, this is expressed as: \[ \sum_{n=1}^{\infty} \frac{1}{n^2} \] The solution to the Basel Problem is known, and evaluating \( \zeta(2) \) provides the exact value. - **Convergence of Series:** - For \( s \) with a real part greater than 1, the series converges, meaning the sum approaches a specific finite value. This property ensures that \( \zeta(s) \) is defined and exists for these values of \( s \). --- This content should assist learners in understanding the practical application of the Riemann zeta function in relation to the Basel Problem and the conditions under which the series converges.
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