Intereting Posts

Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory
Order of the product of two commuting elements with coprime orders in a group.
Original author of an exponential generating function for the Bernoulli numbers?
degree 3 Galois extension of $\mathbb{Q}$ not radical
Derivative of cross-product of two vectors
Can two perfect squares average to a third perfect square?
Notation to work with vector-valued differential forms
Taking trace of vector valued differential forms
Meaning of Normal Vector in Surface Integration
Studying mathematics efficiently
Help finding a combinatorial proof of $k {n \choose k } = n {n – 1 \choose k -1}$
Proving the series of partial sums of $\sin (in)$ is bounded?
Which harmonic numbers have prime numerators?
Compute Angle Between Quaternions (in Matlab)
In what order should the following areas of mathematics be learned?

Is it true that

$$

\left|\sum_{k=1}^n \sin(k) \right|\leq M

$$

for every $n$?

I tried comparing this to the integral $\int_2^\infty \sin(x)dx$ but it is not monotone.

This is part of a problem that says if $A_n=a_1+a_2+\cdots+a_n$ and $|A_n|\leq M$ and $b_n$ is a decreasing sequence to zero, then $\sum a_n b_n$ converges. Then use this to show that

$$

\sum_{n=2}^\infty \frac{\sin(n)}{\log(n)}

$$

- $x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.
- Rigorous proof of an infinite product.
- How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$
- Is this sum related to the Gregory's limit?
- Asymptotics of a summation over real valued functions
- max {$f_n(x):x\in$}$\to$ max{$f(x):x\in$}

- Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$
- Use Fourier series for computing $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$
- Discriminant for $x^n+bx+c$
- How does one evaluate $\lim\limits _{n\to \infty }\left(\prod_{x=2}^{n}\frac{x^3-1}{x^3+1}\right)$?
- What is the family of generating functions for the *rows* of this Stirling-number matrix for whose columns they are $\exp(\exp(x)-1)-1 $?
- Series for envelope of triangle area bisectors
- Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?
- Proof of Divergence for a Sequence
- Integrate $\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$
- To find the limit of three terms mean iteration

- A differentiation under the integral sign
- Winding Numbers and Fixed Point Theorems
- “Honest” introductory real analysis book
- Difference between elementary submodel and elementary substructure
- Proof of derivative of $e^x$ is $e^x$ without using chain rule
- Why does factoring eliminate a hole in the limit?
- How to find a smallest group generator
- How do you find the center of a circle with a pencil and a book?
- Brownian motion introduction
- Field automorphisms of $\mathbb{Q}$ – shouldn't there be only one?
- Fundamental Theorem of Trigonometry
- How to find this determinant of the binomial coefficient $\det{(A)}$
- Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.
- Optimization problem using Reproducing Kernel Hilbert Spaces
- Operator: not closable!