1
vote
1answer
42 views

Why is the Taylor expansion of $\cos$ decreasing?

Why is the Taylor expansion of $\cos$ decreasing ? $\cos(t)=1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+...$ such that one can estimate $\cos(t)<1-\frac{t^2}{2!}+\frac{t^4}{4!}$ I ...
0
votes
0answers
50 views

is there any other method to solve this limit [duplicate]

If someone prove this $$ \lim_{x \to 0}\frac{\sin{x}}{x} =1$$ other than using L'Hospital or series expansion, it will be really appreciable
0
votes
1answer
112 views

Proof that the sum $\sum _{n=1} ^{\infty} (-1)^n \sin (nx)$ is bounded

How can I prove that there is some constant $M>0$, such that for all $N\in\mathbb{N}$ and $x\in [0,\pi]$, $$\left|\sum _{n=1} ^{N} (-1)^n \sin (nx)\right| < M\text{?}$$
3
votes
2answers
254 views

Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in [0,2 \pi]$?

Does the series $$ \sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}, $$ converge for all $\alpha > \frac{1}{2}$ and for all $x \in [0,2 \pi]$? It is obvious that it does when $\alpha > 1$, ...
10
votes
1answer
194 views

Limit of maximum of $f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)})$

let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$ My try: since ...
3
votes
1answer
437 views

Sum of tangent functions where arguments are in specific arithmetic series

By looking through an book, I found this interesting series To prove that: $$\tan(\theta)+\tan \left(\theta+ \frac{\pi}{n} \right) + \tan(\theta + \frac{2\pi}{n}) + \dots + \tan \left (\theta + ...
0
votes
0answers
120 views

Cosine series as a Fourier series.

Theorem: Let $a_{n}\downarrow 0$ and suppose that $\left(a_{n}\right)$ is a quasi-convex. Then $\displaystyle \frac{a_{0}}{2}+\sum a_{n}\cos nx$ is the Fourier series of the $L^{1}$ function ...
0
votes
0answers
135 views

Trigonometric series as a Fourier series of essentially bounded function.

A trigonometric series $ \displaystyle \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx +b_{n}\sin nx)$ is a Fourier series of a essentially bounded function if and only if there exists a constant $K$ ...