Intereting Posts

Solving $(z+1)^5 = z^5$
Almost sure convergence of maximum in a sequence of Gaussian random variables
Infinite derivative
Why the principal components correspond to the eigenvalues?
$R/Ra$ is an injective module over itself
Existence of Gergonne point, without Ceva theorem
Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?
Diophantine quartic equation in four variables, part deux
Are there statements that are undecidable but not provably undecidable
Is this casino promotion exploitable?
Prove that $\Bbb{Z}/I$ is finite where I is an ideal of $\Bbb{Z}$
Stochastic Processes Solution manuals.
If $x \equiv 1 \pmod 3$ and $x \equiv 0 \pmod 2$, what is $x \pmod 6$?
Integral domain with fraction field equal to $\mathbb{R}$
Generating function identity from number of irreducible monic polynomials in $\mathbf{GF}(q)$.

I found a nice general formula for the Taylor series of $\tan x$:

$$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n – 1} $$

where $B_n$ are the Bernoulli numbers and $|x| < \dfrac {\pi} 2$.

- Explain inequality of integrals by taylor expansion
- How to find the Laurent series for $f(z)=\frac{2}{(z-4)}-\frac{3}{(z+1)}$
- Transforming an integral equation using taylor series
- Formula for calculating $\sum_{n=0}^{m}nr^n$
- How Does Integrating the Derivative of this Approximation for Tangent Increase its Accuracy?
- This one weird thing that bugs me about summation and the like

I’ve tried Googling for a proof but didn’t find anything. Hints would be appreciated too.

I am using the typical definition of the Bernoulli numbers:

$$\frac x {e^x – 1} = \sum_{n\,=\,0}^\infty \frac {B_n x^n} {n!}$$

- Do all polynomials with order $> 1$ go to $\pm$ infinity?
- Is there an approximation to the natural log function at large values?
- How can I accurately compute $\sqrt{x + 2} −\sqrt{x}$ when $x$ is large?
- Computing the trigonometric sum $ \sum_{j=1}^{n} \cos(j) $
- Closed form for $\prod_{n=1}^\infty\sqrt{\tanh(2^n)},$
- Can the limit $\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$ be calculated?
- How to expand $\sqrt{x^6+1}$ using Maclaurin's series
- To compute $\tan1-\tan3+\tan5-\cdots+\tan89$, $\tan1+\tan3+\tan5+\cdots+\tan89$
- How to prove $\sum_{k=1}^n \cos(\frac{2 \pi k}{n}) = 0$ for any n>1?
- taylor expansion in cylindrical coordinates

$$\frac z{e^z-1}+\frac z 2=1+\sum_{n=2}^\infty\frac{B_n}{n!}z^n$$

Replace $z$ with $2iz$ to get

$$\color{red}{z\cot(z)}=\frac{iz(e^{iz}+e^{-iz})}{e^{iz}-e^{-iz}}=1+\sum_{n=2}^\infty\frac{B_n}{n!}(2iz)^n=1+\sum_{n=1}^\infty\frac{B_{2n}}{(2n)!}(-1)^n(2z)^{2n}$$

Now use following trigonometric formula

$$\tan(z)=\cot(z)-2\cot(2z).$$

The only non zero $B_n$ with odd index is $B_1=-1/2$, So

$$

\sum_{n=0}^\infty\frac{B_{2n}}{(2n)!}x^{2n}=x\left(\frac{1}{e^x-1}+\frac{1}{2}\right)

=\frac{x}{2}\coth\left(\frac{x}{2}\right)

$$

which is valid for $|x|<2\pi$. Applying this with $x=it$, we get for $|t|<2\pi$:

$$

\sum_{n=0}^\infty\frac{(-1)^nB_{2n}}{(2n)!}t^{2n}= \frac{t}{2}\cot\left(\frac{t}{2}\right)

$$

Now note that

$$

\cot(t)-2\cot(2t)=\frac{1-\cos^2t }{\sin t\cos t}=\tan t

$$

so, for $|t|<\dfrac{\pi}{2}$ we have

$$\eqalign{

t\tan t&=t\cot(t)-2t\cot(2t)\cr

&=\sum_{n=0}^\infty\frac{(-1)^nB_{2n}2^{2n}}{(2n)!}t^{2n}

-\sum_{n=0}^\infty\frac{(-1)^nB_{2n}4^{2n}}{(2n)!}t^{2n}\cr

&=\sum_{n=1}^\infty\frac{(-1)^nB_{2n}2^{2n}(1-2^{2n})}{(2n)!}t^{2n}\cr

&=t\sum_{n=1}^\infty\frac{(-1)^{n-1}B_{2n}2^{2n}(2^{2n}-1)}{(2n)!}t^{2n-1}

}

$$

From this the desired expansion follows.$\qquad\square$.

- Sum of divergent series
- What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?
- Bellard's exotic formula for $\pi$
- Deriving the Bring-Jerrard quintic using a $cubic$ Tschirnhausen transformation
- half space is not homeomorphic to euclidean space
- How prove this mathematical analysis by zorich from page 233
- Weak convergence and strong convergence
- What's $(-1)^{2/3}\; $?
- Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences
- Insightful proofs for Sherman-Morrison Formula and Matrix Determinant Lemma
- On clarifying the relationship between distribution functions in measure theory and probability theory
- Topological groups are completely regular
- Topology on the space of test functions
- Bounded sequence and every convergent subsequence converges to L
- Poincaré lemma on a space with trivial homology group