Intereting Posts

Inequalities on kernels of compact operators
Prove by induction. How do I prove $\sum\limits_{i=0}^n2^i=2^{n+1}-1$ with induction?
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$
Monotone Convergence Theorem – clarification on measurable set.
Existence of uncountable set of uncountable disjoint subsets of uncountable set
Partial derivatives and orthogonality with polar-coordinates
an analytic function from unit disk to unit disk with two fixed point
Very elementary proof of that Euler's totient function is multiplicative
Is there anything like “cubic formula”?
Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$
Prove $\exists T\in\mathfrak{L}(V,W)$ s.t. $\text{null}(T)=U$ iff $\text{dim}(U)\geq\text{dim}(V)-\text{dim}(W)$.
Are there arbitrarily large sets $S$ of natural integers such that the difference of each pair is their GCD?
Converging series question, Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then does $\sum_{n=1}^{\infty} \frac {a_n}{n}$
Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K
Osgood condition

Here’s the problem:

Given $|f(x)| = 1$, construct an $f(x)$, such that $$\int ^{+\infty }_{0}f\left( x\right) dx$$ converges.

I think this problem may be done by dividing the 1s and -1s smartly, but I haven’t got any workable ideas on it.

- Difference between a Gradient and Tangent
- composition of continuous functions
- An alternative proof for sum of alternating series evaluates to $\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$
- What is the significance of the three nonzero requirements in the $\varepsilon-\delta$ definition of the limit?
- How to obtain the series of the common elementary functions without using derivatives?
- Does an absolutely integrable function tend to $0$ as its argument tends to infinity?

- p-series convergence
- Geometric Interpretation of Total Derivative?
- Exact value of $\sum_{n=1}^\infty \frac{1}{n(n+k)(n+l)}$ for $k \in \Bbb{N}-\{0\}$ and $l \in \Bbb{N}-\{0,k\}$
- Is the Nested Radical Constant rational or irrational?
- Given $a_{1}=1, \ a_{n+1}=a_{n}+\frac{1}{a_{n}}$, find $\lim \limits_{n\to\infty}\frac{a_{n}}{n}$
- How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$
- Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$
- Multivariable calculus - Implicit function theorem
- Is my Riemann Sum correct?
- Why does the sum of these trigonometric expressions give such a simple result?

Rather than giving you an integral that converges, I’ll give you an example of a divergent integral that comes from a divergent series.

(Just adopt the method I am showing you to a convergent series)

$1-2+3-4+5-….$ is a divergent series.

I am going to construct an integral that is “equal to” this series.

$f(x) = \begin{cases}

1, &[0,1]\\

-1, &[1,3]\\

1, &[3,6]\\

-1, &[6,10]\\

\dots

\end{cases}$

Can you see that

$\int_0^\infty f(x) dx = 1 – 2 + 3 – 4 +\dots$?

Do you see how you can adopt this for any (convergent or divergent) series?

A classic example is something like $\int_0^\infty e^{ix^2}\;dx$. Letting $x=\sqrt{y}$ gives a sort of alternating-decreasing situation whose convergence becomes more obvious.

- Exercise 1 pg 33 from “Algebra – T. W. Hungerford” – example for a semigroup-hom but not monoid-hom..
- Must the (continuous) image of a null set be null?
- Showing that complex exponentials of the Fourier Series are an orthonormal basis
- Evaluating the Poisson Kernel in the upper half space in $n$-dimensions
- Explain why $x^+=A^+b$ is the shortest possible solution to $A^TA\hat{x}=A^Tb$
- Pigeonhole principle for a triangle
- How prove this $(abc)^4+abc(a^3c^2+b^3a^2+c^3b^2)\le 4$
- Examples of categories where epimorphism does not have a right inverse, not surjective
- Combinatorial Proof Of ${n \choose k}={n-1\choose {k-1}}+{n-1\choose k}$
- Convergence of alternating series based on prime numbers
- a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$
- Find the limit of $(\sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdot … \cdot \sin 1)^{\frac{1}{n}}$
- Intervals are connected and the only connected sets in $\mathbb{R}$
- Translate of an ideal by $a$ coincides with the ideal iff $a \in I$
- Determine splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 – 2$