Intereting Posts

REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
How Do I Compute the Eigenvalues of a Small Matrix?
Free Throw Probability and Expected Number of Points
Finding Eccentricity from the rotating ellipse formula
Maximal finite order of Abelian Groups
Does anyone know a good hyperbolic geometry software program?
Fibonacci Generating Function of a Complex Variable
Proving two measures of Borel sigma-algebra are equal
please solve a 2013 th derivative question?
Proofs from the BOOK: Bertrand's postulate Part 3: $\frac{2}{3}n<p \leq n \rightarrow$ no p divides $\binom{2n}{n}$
exterior covariant derivative of $\operatorname{End}(E)$-valued $p$-form
Is every irrational number normal in at least one base?
Convergence of a series $\sum\limits_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\frac{1}{2}$
relation between integral and summation
Number of permutations with a given partition of cycle sizes

How to prove $$\sum_{k=1}^{\infty} \frac{\sin(kx)}{k}$$ converges without using integral test?

- Showing that $\frac{\sqrt{n!}}{n}$ $\rightarrow \frac{1}{e}$
- Limit of CES function as $p$ goes to $- \infty$
- Limit of the composition of two functions with f not necessarily being continuous.
- If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
- Limit of $n/\ln(n)$ without L'Hôpital's rule
- Series which are not Fourier Series
- Bounding or evaluating an integral limit
- Equicontinuity on a compact metric space turns pointwise to uniform convergence
- Prove that the series $\sum_{n=1}^\infty \left$ converges
- Definition of the Limit of a Function for the Extended Reals

This follows from Dirchlet’s Test. We identify $a_n = \frac{1}{n}$ and $b_n = \sin(nx)$ as in the linked theorem.

We are left to check the three properties

$(a_n)$ is monotonic decreasing, and is bounded away from $0$.

$a_n \to 0$ as $n \to \infty$.

$\left | \sum_{n < N} b_n \right | \leq M$ for each $N$.

I think (1) and (2) are straightforward to see. For (3), notice that $$\sum_{n = 1}^{N} 2\sin(nx)\sin \left ( \frac{x}{2} \right ) = \sum_{n = 1}^N \cos \left ( n-\frac{1}{2} \right) x – \cos\left (n + \frac{1}{2} \right)x.$$

The sum on the right is telescoping, so dividing both sides by $2\sin \left ( \frac{x}{2} \right )$ gives $$\sum_{n = 1}^{N}\sin(nx) = \frac{\cos \frac{x}{2}- \cos\left (N + \frac{1}{2} \right)x.}{2\sin \left ( \frac{x}{2} \right )}$$

I will let you draw the rest of the conclusions.

- Choosing parametrization for complex integration with two branch cuts
- What is the lowest positive integer multiple of $7$ that is also a power of $2$ (if one exists)?
- Two finite abelian groups with the same number of elements of any order are isomorphic
- Intuition about the second isomorphism theorem
- Algebraically-nice general solution for last step of Gaussian elimination to Smith Normal Form?
- Existence of the absolute value of the limit implies that either $f \ $ or $\bar{f} \ $ is complex-differentiable
- What's wrong with this argument? (Limits)
- Number of spanning trees in a ladder graph
- Fluids Euler's Equation
- If $a=\langle12,5\rangle$ and $b=\langle6,8\rangle$, give orthogonal vectors $u_1$ and $u_2$ that $u_1$ lies on a and $u_1+u_2=b$
- Sum of compact sets
- Second-order non-linear ODE
- Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime.
- Basis for a product Hilbert space.
- Book on arithmetic and elementary number theory