Why is this identity true? $$\sum_{k=1}^{\infty} \frac{\sin(k\pi/2)}{k^2} = G$$ where $G$ is Catalan's constant.
2025-01-13 02:36:01.1736735761
Sum of infinite series - Catalan constant
304 Views Asked by T. Böhm https://math.techqa.club/user/t-bohm/detail At
1
There are 1 best solutions below
Related Questions in SEQUENCES-AND-SERIES
- Series to infinity
- Proving whether the limit of a sequence will always converge to 0?
- How come pi is in this question?
- finding limit equation of a convergent sequence (vn) in math exercise
- Convergence of difference of series
- Proof using triangle inequality
- sum of digits = sum of factors
- My incorrect approach solving this limit. What am I missing?
- Using the Monotone Convergence Theorem to prove convergence of a recursively defined sequence.
- Difference of Riemann sums
Related Questions in CONSTANTS
- Find the constants to make it continuous.
- Irrationality of $\pi+c$
- Show that $\lim_{n \to \infty} p(n) = 1-\frac{1}{e}$
- Taking derivatives of constants and variables
- Another way to express $\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$?
- Why is $e$ the number that it is?
- Which continued fraction for $e$ is the most computationally efficient?
- Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?
- What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?
- Sum of infinite series - Catalan constant
Related Questions in CATALANS-CONSTANT
- How to calculate this infinite sum?
- Formulae for Catalan's constant.
- How to evaluate this integral $\int_{0}^{\infty }\frac{\ln\left ( 1+x^{3} \right )}{1+x^{2}}\mathrm{d}x$
- How to evaluate $\int_{0}^{1}\frac{\arctan x}{1+x^{2}}\ln\left ( \frac{1+x^{2}}{1+x} \right )\mathrm{d}x$
- Catalan's constant and $\int_{0}^{2 \pi} \int_{0}^{2 \pi} \ln(\cos^{2} \theta + \cos^{2} \phi) ~d \theta~ d \phi$
- $\int_0^{\pi/2} \sec^a(t)\,dt= \frac{\sqrt{\pi}}{2\Gamma\left(1-\frac{a}{2}\right)}\Gamma\left(\dfrac{1-a}{2}\right)$
- Integral $\int_{0}^{\frac{\pi}{4}}x\ln(\cos x)dx$
- Riemann Zeta function and Catalan's constant
- How far have we got towards proving Catalan's constant is irrational?
- A double sum identity for Catalan constant
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
The series sums to $$\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\cdots$$ which is just the definition of Catalan's constant.