Intereting Posts

Why is negative times negative = positive?
What number appears most often in an $n \times n$ multiplication table?
Does the golden angle produce maximally distant divisions of a circle?
Why does an integral change signs when flipping the boundaries?
Finite index subgroups of a virtually abelian group
Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$
$\sum_{m=1}^{\infty}{\frac{e^{-a m^2}}{m^2}}$ as an Integral
What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?
In a group of 6 people either we have 3 mutual friends or 3 mutual enemies. In a room of n people?
Exponential Generating Functions For Derangements
Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)
Difference between functional and function.
Ordered tuples of proper classes
Is $G/pG$ is a $p$-group?
One Question about the Fubini's Theorem

We have to show that

$$ \lim_{n\to\infty} \left( \dfrac 1 {\sqrt{n^2 +1}} +\dfrac 1 {\sqrt{n^2 +2}} + \dfrac 1 {\sqrt{n^2 +3}} + \ldots + \dfrac 1 {\sqrt{n^2 +n}} \right) = 1$$

Now, I thought of doing it as a definite integral, however the terms don’t translate into a function of $\left(\dfrac r n\right)$. I tried searching wolfram-alpha, but it used Hurwitz-zeta function, of which I have no clue. (This is an assignment problem)

- Finding parametric curves on a sphere
- The Limit of $x\left(\sqrt{a}-1\right)$ as $x\to\infty$.
- Calculating line integrals via Stokes theorem
- What is Jacobian Matrix?
- Find for which value of the parameter $k$ a function is bijective
- How to compute the following definite integral

- Show that: $\lim\limits_{n\to\infty} \sqrtn = 1$
- Monotonic subsequences and convergence
- How to prove that exponential grows faster than polynomial?
- Interpreting the significance of Darboux's Theorem
- Show that $\int_0^\infty \frac{x\log(1+x^2)}{e^{2\pi x}+1}dx=\frac{19}{24} - \frac{23}{24}\log 2 - \frac12\log A$
- $ \int_{0}^{ \infty} \int_{0}^{ \infty} \frac { e^{-(x+y)}}{x+y} dx dy $
- Proof that a degree 4 polynomial has at least two roots
- Spivaks Chain Rule Proof (Image of proof provided)
- Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge?
- Compute $\lim_{x\to\infty}x\;\left$

**Hint:** For all $n\in \mathbb N$ it holds that

$$ \dfrac n {\sqrt{n^2 +n}} \leq \dfrac 1 {\sqrt{n^2 +1}} +\dfrac 1 {\sqrt{n^2 +2}} + \dfrac 1 {\sqrt{n^2 +3}} + \ldots + \dfrac 1 {\sqrt{n^2 +n}}\leq \dfrac n {\sqrt{n^2 +1}}.$$

- Continuous Mapping Theorem for Random Variables
- Examples of Non-Noetherian Valuation Rings
- Proof of properties of dual cone
- $f_n(x) = x – x^n$ for $x\in $. Does the sequence converge pointwise or uniformly on $$?
- Eigenvalue Problem — prove eigenvalue for $A^2 + I$
- $G$ be a non-measurable subgroup of $(\mathbb R,+)$ ; $I$ be a bounded interval , then $m^*(G \cap I)=m^*(I)$?
- What is the asymptotic behavior of a linear recurrence relation (equiv: rational g.f.)?
- Why is $^\mathbb{N}$ not countably compact with the uniform topology?
- Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$
- Uniform continuity question
- How exactly is $i=\sqrt{-1}$ related to $\mathbb{C}$ being a closed algebraic field?
- A question on coalgebras(1)
- What's special about the greatest common divisor of a + b and a – b?
- Relations of a~b iff b = ak^2
- Can a function have a strict local extremum at each point?