Intereting Posts

Eisenstein's Criterion stronger version
Prove: If $n=2^k-1$, then $\binom{n}{i}$ is odd for $0\leq i\leq n$
Using Taylor expansion to find $\lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1}$
Numer of solutions for IVP
An Identity Involving Narayana Numbers
Convergence of “alternating” harmonic series where sign is +, –, +++, —-, etc.
What are functions used for?
A probability function is determined on a dense set- Where is density used in the following proof?
Prove that a monomial ideal $I$ is determined by the set of monomials it contains.
Derivation of the polarization identities?
Teenager solves Newton dynamics problem – where is the paper?
How to prove $\int_0^1 \frac{1+x^{30}}{1+x^{60}} dx = 1 + \frac{c}{31}$, where $0 < c < 1$
Infinite nilpotent group, any normal subgroup intersects the center nontrivially
Using substitution while using taylor expansion
Prime factorization of $\frac{3^{41} -1}{2}$

The following is a homework problem:

Prove that between two roots of the function $f(x) = \exp(x) \sin(x) -1$ there must be at least one root of the function $\exp(x)\cos(x) +1$.

I think it is possible to prove it by simply computing the signs of the functions at $\frac{\pi}{2}k, \pi k, \frac{3\pi}{2}k, 2\pi k$. However, I’m looking for a more elegant way. It is intuitively clear that the roots of $f$ are ˋˋconverging” to the roots of $sin$ and the ones of $g$ to the ones of $cos$. Do you have any hint how to construct a nice argument based on this observation?

- Is $$ a countable disjoint union of closed sets?
- How to prove this function is integrable??
- Infinite series $\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$
- Some help needed to evaluate the following integral using Residue theorem
- The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets
- A necessary condition for a multi-complex-variable holomorphic function.

- Given point and tangent line
- Finding the convergence interval of $\sum\limits_{n=0}^{\infty} \frac{n!x^n}{n^n}$.
- What does smooth curve mean?
- Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional
- Is the limit finite? (corrected)
- Solving a Word Problem relating to factorisation
- $ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $
- What is an intermediate definition for a tangent to a curve?
- Prove $\int_{0}^{\pi/2} x\csc^2(x)\arctan \left(\alpha \tan x\right)\, dx = \frac{\pi}{2}\left$
- Weird and difficult integral: $\sqrt{1+\frac{1}{3x}} \, dx$

The functions $g(x)=\sin(x)-\exp(-x)$ and $g'(x)=\cos(x)+\exp(-x)$ have the same roots as the given functions. Thus the problem reduces to the theorem of Rolle.

- Rudin Theorem 3.27
- How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?
- Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers.
- How do I calculate the intersection(s) of a straight line and a circle?
- Evaluating the series $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $
- Order of a product in an abelian group.
- Smallest function whose inverse converges
- Can the entropy of a random variable with countably many outcomes be infinite?
- Perfect Square relationship with no solutions
- Elementary solution of exponential Diophantine equation $2^x – 3^y = 7$.
- Why does this covariance matrix have additional symmetry along the anti-diagonals?
- Why is pointwise continuity not useful in a general topological space?
- How to convert an English sentence that contains “Exactly two” or “Atleast two” into predicate calculus sentence?
- Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$
- How to find the smallest $n$ such that $n^a\equiv 1 \pmod p$