4. The Reimann zeta-function is defined by II. and is used in number theory to study the distribution of primes. (a) For values of x does ¢ converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4 n=1 $(2) = 1 -£-/-/- n=1 1 (n + 2)² 4 3 (³)* = π² ∞ 1 n² 6 ζ(x) = Σ Use Euler's results to find the sum of the following series: ∞ ∞ 1. Σ 1 III. Σ n=3 (n − 2)4 - n=5 n=j nữ ∞ IV. Σ 1 ∞ ζ(4) = Σ n=1 n=2 n² 1 TA 90 n4 =
4. The Reimann zeta-function is defined by II. and is used in number theory to study the distribution of primes. (a) For values of x does ¢ converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4 n=1 $(2) = 1 -£-/-/- n=1 1 (n + 2)² 4 3 (³)* = π² ∞ 1 n² 6 ζ(x) = Σ Use Euler's results to find the sum of the following series: ∞ ∞ 1. Σ 1 III. Σ n=3 (n − 2)4 - n=5 n=j nữ ∞ IV. Σ 1 ∞ ζ(4) = Σ n=1 n=2 n² 1 TA 90 n4 =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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Transcribed Image Text:4. The Reimann zeta-function ζ is defined by
and is used in number theory to study the distribution of primes.
(a) For values of x does ( converge? (i.e What is the domain of (?) Explain your reasoning.
(b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4
0
1. Σ
n=3
Ç(2) =
1
(n + 2)2
II.
1. Σ(3)
n=1
00
4
n=1
π
Use Euler's results to find the sum of the following series:
1
(n - 2)4
00
ς(α)
4(x) = Ση
n=1
2
0
III. Σ
η=5
IV. Σ
n=2
$(4) =
1
η2
00
n=1
=
.4
η4 90
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