# Tagged Questions

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### Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2}$, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
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### Find the limit of $\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$

Find the limit of $$\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$$ The limit does exist, but I can not get it. Thanks Willie-Wong & Lee Mosher for correcting the expression.
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### Can somebody explain to me why these terms are equal?

I just read a proof on ProofWiki that proves Euler's formula, but I can't seem to understand what is done in this following step: ...
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### Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
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### Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Please help me to find a closed form for the infinite product $$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$ where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
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### Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$

I need help with calculating this sum: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
If you have $\sum_{n = 0}^\infty(4/5)^n$ and you are asked to represent it as a geometric series you would: $\sum_{n = 0}^\infty(4/5)(4/5)^{n-1}$ //factor out your constant therefore $a = 4/5$, ...
My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...