Intereting Posts

How to pass from $L^2(0,T;V')$ to $\mathcal{D}'\big(\Omega\times (0,T)\big)$?
Moscow puzzle. Number lattice and number rearrangement. Quicker solution?
Prove that integral of continuous function is continuously differentiable
Solving a first order linear ODE and determining the behavior of its solutions
Convergence of $\sum^\infty_{n=1} \ln(1+\frac 1 {2^n})$
If $A$ and $B$ are nonempty sets, prove that $A \times B = B \times A$ if and only if $A = B$
Equivalence for Christoffel symbol and Koszul formula
Why are there no discrete zero sets of a polynomial in two complex variables?
Calculating $\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz$
Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$
$M$ is a compact manifold with boundary $N$,then $M$ can't retract onto $N$.
For which $\mathcal{F} \subset C$ does there exist a sequence converging pointwise to the supremum?
Possible to solve this differential equation?
$ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $
Combinatorial interpretation of Binomial Inversion

How to determine if the following series are convergent or divergent? I’m supposed to use here the limit comparison test, but I don’t know how to choose the second series.

$$\sum_{k=1}^\infty \ln(1+ \sqrt{\frac 2k})$$

$$\sum_{k=1}^\infty\displaystyle \sqrt[k]{e}\sin(\frac{\pi}{k}).$$

- Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
- R as a union of a zero measure set and a meager set
- Is there any integral for the Golden Ratio?
- Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.
- Convergence of a sequence (possibly Riemann sum)
- How do telescopic series work in general and in this specific problem?
- A strange “pattern” in the continued fraction convergents of pi?
- Finding cubic function from points?
- When is a sequence $(x_n) \subset $ dense in $$?
- Prove the convergence/divergence of $\sum \limits_{k=1}^{\infty} \frac{\tan(k)}{k}$

A related problem. Since

$$ \lim_{k\to \infty }\frac{\ln(1+\sqrt{2/k})}{\sqrt{2/k}}=1, $$

then the series diverges by the fact:

Suppose $\sum_{n} a_n$ and $\sum_n b_n $ are series with positive terms, then

if $\lim_{n\to \infty} \frac{a_n}{b_n}=c>0$, then either both series converge or diverge.

**Note:** We used the Taylor series

$$ \ln(1+t)=t+O(t^2)\implies \ln(1+t)\sim t. $$

$$e^t\sin(\pi t)= \pi t+O(t^2)\implies e^t\sin(\pi t) \sim \pi t.$$

- $(-32)^{\frac{2}{10}}\neq(-32)^{\frac{1}{5}}$?
- Is $B = A^2 + A – 6I$ invertible when $A^2 + 2A = 3I$?
- Intermediate value property problem and continuous function
- Why is $Sp(2m)$ as regular set of $f(A)=A^tJA-J$, and, hence a Lie group.
- Building a hidden markov model with an absorbing state.
- advantage of first-order logic over second-order logic
- $\mathbb{Z}$ is the symmetry group of what?
- Finding a invariant subspaces for a specific matrix
- tensor product of sheaves commutes with inverse image
- Is it true that any matrix can be decomposed into product of rotation,reflection,shear,scaling and projection matrices?
- Does almost sure convergence implies convergence of the mean?
- A group of order $108$ has a proper normal subgroup of order $\geq 6$.
- Collatz conjecture: Largest number in sequence with starting number n
- Can it be determined that the sum of the diagonal entries, of matrix A, equals the sum of eigenvalues of A
- Elementary number theory – prerequisites