Q1: The sum of the infinite series $\cot ^{-1}2 + \cot ^{-1} 8+ \cot^{-1}18+ \cot^{-1}32\cdots$ 1.$\pi/3$ 2.$\pi/4$ 3.$\pi/2$ 4.None Q2: Value of $\lim_ {n \to \infty}[ {\cos \frac{\pi}{2^2} } {\cos \frac{\pi}{2^3} } \ldots{\cos \frac{\pi}{2^n} }$] $\pi$ $1/\pi$ $2/\pi$ $\pi/e$

Use $cos (n\theta)$ = $\frac{z^n +z^{-n}}{2}$ to express $\cos \theta + \cos 3\theta + \cos5\theta + … + \cos(2n-1)\theta$ as a geometric series in terms of z. Hence find this sum in terms of $\theta$. I’ve tried everything in the world and still can’t match that of the final answer. Could I pleas have a […]

Find the value of $\dfrac{\sum_\limits{k=0}^{6}\csc^2\left(x+\dfrac{k\pi}{7}\right)}{7\csc^2(7x)}$ when $x=\dfrac{\pi}{8}$. The Hint given is: $n\cot nx=\sum_\limits{k=0}^{n-1}\cot\left(x+\dfrac{k\pi}{n}\right)$ I dont know how it comes nor how to use it

Question: If $x \neq 0$, then prove that $\displaystyle \sum_{n=1}^{\infty}\dfrac1{2^n} \tan\left(\dfrac{x}{2^n}\right) = \dfrac1{x} – \cot x.$ My answer: I proved this result by using the following identity: $$ \prod_{k=1}^n \cos\left(\frac{x}{2^k}\right) = \frac{\sin x}{2^n\sin \frac{x}{2^n}}$$ I took natural log on both sides of the above equation and then differentiated both sides to get $$\sum_{k=1}^n \frac1{2^k} \tan\left(\frac{x}{2^k}\right) […]

de Moivre’s identity $$ (\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta $$ only applies as written when $n \in \mathbb{Z}$. if the exponent is a fraction $\frac{m}{n}$ then there will be $n$ values of $(\cos \theta + i \sin \theta)^{\frac{m}{n}}$, whilst for irrational $\alpha$ the set $\{e^{i\alpha (\theta +2\pi […]

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I’ve tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of $\cos(2k-1)x$) but It has many roots. So, I couldn’t go further.

I am studying Furstenberg’s article Strict ergodicty and transformation of the torus and I’m stuck with the following construction. Define sequence $(v_k)_{k \in \mathbb{N}}$ as $v_1 =1, v_{k+1}=2^{v_k}+v_k+1$ and sequence $(n_k)_{k \in \mathbb{Z} \backslash \{0\}}$ as $n_k=2^{v_k}$ for $k>0$ and $n_k = – n_{-k}$ for $k<0$. Consider function $H(\theta) : [0,1) \rightarrow \mathbb{R}$ given by […]

I’m taking a single variable calculus course and sine series is defined as $$ \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k + 1)!}. $$ Sine is also defined as opposite length over hypotenuse length for a right angled triangle. So $$\frac{\text{opposite}}{\text{hypotenuse}} = \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k + 1)!}? $$ Is there a concept I’m not understanding here?

I’ve noticed the derivative of $\sin^2 x$, $$\frac{d}{dx}\sin^2(x)=\sin 2x$$ and the same applies to $\sinh^2 x$, that is $$\frac{d}{dx}\sinh^2(x)=\sinh(2x)$$ I was wondering about the other functions of that property. (I posted a similar question about a week ago but I had an incorrect premise.) So these functions must satisfy $$(f^2(x))’=f(2x)\rightarrow 2f'(x)f(x)=f(2x)$$ The only way I […]

How do we compute : $$i)\ S_1 = \tan1-\tan3+\tan5-\cdots+\tan89$$ and $$ii)\ S_2 = \tan1+\tan3+\tan5+\cdots+\tan89$$ all the angles are in degrees. Thanks

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