Intereting Posts

How do you calculate this limit?
Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”
A net version of dominated convergence?
Calculate integrals involving gamma function
Positive Linear Functionals on Von Neumann Algebras
Power Series proofs
Do rotations about any three non-collinear axes generate $SO(3)$?
Maximal and Prime ideals of cross product of rings
Bounded (from below) continuous local martingale is a supermartingale
Methods for showing three points in $\mathbb{R}^2$ are colinear (or not)
Proof That $\mathbb{R} \setminus \mathbb{Q}$ Is Not an $F_{\sigma}$ Set
Finding a closed form for $\cos{x}+\cos{3x}+\cos{5x}+\cdots+\cos{(2n-1)x}$
A proof for Landau inequality and similar cases
How to show that Klein four-group is a normal subgroup of the alternating group $A_4$
When is the quotient space of a second countable space second countable?

$$\sum_{n=0}^\infty \frac{\tan(a/2^n)}{2^n},$$

where $a$ isn’t a multiple of $\pi$. I’ve been going through several telescoping questions, and It seems I have hit a brick wall with this one, any help will be appreciated.

- A functional relation which is satisfied by $\cos x$ and $\sin x$
- Seemingly invalid step in the proof of $\frac{a^2+b^2}{ab+1}$ is a perfect square?
- “If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”
- Compositeness of $n^4+4^n$
- Solve $\lim_{x\to +\infty}\frac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$
- Straightedge-only construction of a perpendicular

- Problem with infinite product using iterating of a function: $ \exp(x) = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x) \cdot \ldots $
- convergence of alternating series — weakening a hypothesis
- What's the value of this Viète-style product involving the golden ratio?
- Prove $ \sin x + \frac{ \sin3x }{3} + … + \frac{ \sin((2n-1)x) }{2n-1} >0 $
- How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$
- Filling an array(Putnam)
- Proving that the limit of a sequence is $> 0$
- Combining stones into one stack
- Arithmetic-geometric mean of 3 numbers
- Can the entropy of a random variable with countably many outcomes be infinite?

Since

$$\tan\left(\frac{a}{2^n}\right)\sim_\infty\frac{a}{2^n}$$

then

$$ \frac{\tan(a/2^n)}{2^n}\sim_\infty\frac{a}{4^n}$$

so the given series is convergent.

**Added** *(If you ask for the sum)* We have

$$\tan t= \frac{1}{\tan t}-\frac{2}{\tan (2t)}$$

hence we find

$$\frac{\tan(a/2^n)}{2^n}= \frac{1}{2^n\tan (a/2^n)}-\frac{1}{2^{n-1}\tan (a/2^{n-1})}=u_n-u_{n-1}$$

where

$$u_n=\frac{1}{2^n\tan (a/2^n)}$$

so by telescoping we have

$$\sum_{n=0}^\infty \frac{\tan(a/2^n)}{2^n}=\lim_{n\to\infty}u_n-u_{-1}=\frac{1}{\tan a}- \frac{2}{\tan (2a)}=\tan a$$

- Kepler, cartesian coordinates and ellipses
- Number of elements in cartesian power with a majority constraint
- what are the product and coproduct in the category of topological groups
- Is $e^x$ the only isomorphism between the groups $(\mathbb{R},+)$ and $(\mathbb{R}_{> 0},*)$?
- Maximal normal $\pi$-subgroups and torsion subgroups
- Solve Burgers' Equation with side condition.
- Using tan(x), show that open interval is diffeomorphic with the real line
- Why does Strassen's algorithm work for $2\times 2$ matrices only when the number of multiplications is $7$?
- Does the functional equation $p(x^2)=p(x)p(x+1)$ have a combinatorial interpretation?
- Are all computable functions continuous or vice-versa?
- Possible all-Pentagon Polyhedra
- Cup product of cohomology and the Kunneth formula
- Non-trivial open dense subset of $\mathbb{R}$.
- Is most of mathematics independent of set theory?
- Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.