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1 vote

Proving $\lim_{n \to \infty} \frac{n}{n+2} \sum_{k=1}^{n} \frac{k}{k+3} = \infty$

While your solution is correct, I think you are complicating things unnecessarily. You could simply bound $\frac{k}{k+3}$ by $\frac{1}{4}$ and thus have $\frac{n}{n+2} \sum_{k=1}^n \frac{k}{k+3} > \...
Kon-kon's user avatar
  • 19
0 votes

Using the binomial expansion to find derivative formulas

There are probably many sources on the internet. Your question isn't very specific and many such formulas occur as exercises. Here is one source: https://mathweb.ucsd.edu/~gptesler/184a/slides/...
Marius S.L.'s user avatar
  • 2,235
0 votes

A question about a multivariable limit

Let's take a line passing through the limit point in parametric form: \begin{align} x &= 1 + l t, \\ y &= 0 + m t \end{align} and substitute into the function $$ F(t) = \frac{l m^2 t}{l^2+3 m^...
Vincenzo Tibullo's user avatar
1 vote

Alternative ways to evaluate $\sum_{k=1}^{n}(2k+1)^2$

You can also make use of generating functions, although this particular method is cumbersome: $$\begin{aligned} \sum_{k=1}^n(2k+1)^2&=\left[\sum_{k=1}^n(2k+1)^2x^{2k+1}\right|_{x=1}\\ &=\left[...
Joan S. Guillamet F.'s user avatar
1 vote

for which p is the infinite series of sin(1/n^p) convergent?

okay I figured myself: since $\frac{1}{n^p}$ tends to 0 we can rewrite $ \sum_{n=1}^{\infty} \sin\frac{1}{n^p}$ as: $$ \sum_{n=1}^{\infty} \frac{1}{n^p}×\frac{\sin\frac{1}{n^p}}{\frac{1}{n^p}} $$ and ...
Delta Account's user avatar
1 vote

Alternative ways to evaluate $\sum_{k=1}^{n}(2k+1)^2$

We can use the generating function to find the sum. Let's consider $$ f(\alpha)=\sum_{k=1}^ne^{-\alpha(2k+1)}=e^{-\alpha}\frac{e^{-2\alpha}-e^{-2\alpha(n+1)}}{1-e^{-2\alpha}}=e^{-\alpha}\frac{1-e^{-2\...
Svyatoslav's user avatar
  • 15.7k
1 vote

Alternative ways to evaluate $\sum_{k=1}^{n}(2k+1)^2$

One pleasant alternative (although one can always quibble about how "alternative" it is) is to exploit the calculus of finite differences. This won't take long: write out the sequence of ...
user43208's user avatar
  • 8,424
3 votes

Alternative ways to evaluate $\sum_{k=1}^{n}(2k+1)^2$

Your alternative ideas don't work indeed. What you can do instead is $$\sum_{k=1}^n(2k+1)^2=\sum_{j=2}^{2n+1}j^2-\sum_{i=1}^n(2i)^2$$ $$=\frac{(2n+1)(2n+2)(4n+3)}6-1-4\frac{n(n+1)(2n+1)}6$$ $$=\frac{n(...
Anne Bauval's user avatar
  • 35.2k
-1 votes

Suppose that $f$ satisfies $\lim_{x \to \infty} x^2 f(x) = 1$. Prove that $\int_{1}^{\infty} f(x) \, dx$ is convergent.

Note that your integral equals $\sum_{k=1}^\infty a_k$ where $a_k=\int_{k}^{k+1} f(x) dx$. For large enough $k$, due to the limit, $x^2 f(x)\le 2$ for all $x\in [k,k+1]$, so $f(x)\le 2/k^2$ and hence $...
van der Wolf's user avatar
  • 2,325
2 votes
Accepted

Suppose that $f$ satisfies $\lim_{x \to \infty} x^2 f(x) = 1$. Prove that $\int_{1}^{\infty} f(x) \, dx$ is convergent.

As $\displaystyle\lim_{x\to\infty} x^2f(x)=1$, there exists some $x_0$ such that $x^2f(x)\leq 2$, or equivalently, $f(x)\leq\dfrac{2}{x^2}$ for $x\geq x_0$. As $$\displaystyle\int_{x_0}^{\infty} \...
Julio Puerta's user avatar
  • 4,311
1 vote

Assistance on an $ε-δ$ proof as $x→∞$

As long as you make sure that $M$ is simultaneously larger than $\frac 1\epsilon$ and also larger than $\frac{10}3$, rather than just larger than $\frac1\epsilon$. This is required for the leftmost ...
Arthur's user avatar
  • 200k
0 votes
Accepted

Calculating the order of $\int_0^r (-\ln x)^{-\gamma} dx$ for $0< \gamma <1$.

Integrating by parts $$\int_0^r 1\times(-\ln(x))^{-\gamma}dx=\left[\frac{x}{(-\ln(x))^\gamma}\right]_0^r-\gamma\int_0^{r}\frac{1}{(-\ln(x))^{\gamma+1}}dx$$ Since $x\mapsto \frac{1}{(-\ln(x))^\gamma}$ ...
Ayoub's user avatar
  • 1,491
3 votes

Exponential equation has no solutions

Assuming this is some Olympiad type problem, a solution using derivatives might not be satisfactory. Clearly, the equation does not have negative solutions, because then the left-hand side is at most ...
naiht's user avatar
  • 51
2 votes

Find number of solutions to the equation $\sin(6\sin x)=\frac{x}{6}$.

If $x$ is a root of the equation, so is $-x$. Furthermore, $x=0$ is a root. For these reasons, we only enumerate the number of positive roots. Since $\frac{x}{6}>1$ for $x>6$, for every positive ...
Mostafa Ayaz's user avatar
  • 32.2k
1 vote

Calculating the order of $\int_0^r (-\ln x)^{-\gamma} dx$ for $0< \gamma <1$.

Performing the change of integration variables from $x$ to $t$ via $x=\mathrm{e}^{-t}$ and utilising the known asymptotic expansion of the generalised exponential integral, we obtain $$ \int_{ - \ln r}...
Gary's user avatar
  • 32.1k
-1 votes

Proving $\lim_{x\to\infty} \left(1+\frac{c}{x}\right)^x = \lim_{x\to\infty}\left(1+\frac1x\right)^{cx} $

To prove the equality of the two limits, we can use the property of exponents which states that $(ab)c = a^{bc}$. Applying this property to the limit, we get: $x→∞lim(1+xc)x=x→∞lim((1+x1)x)c$ Now, the ...
StudyME's user avatar
  • 33
1 vote

Find the set of values of $\alpha$ so that $f(x)=\dfrac{\alpha x^2+6x-8}{\alpha+6x-8x^2}$ is one one.

For $g(x)=-6x+8x^2$ and $h(x)=x^{-1}(\alpha -g(x))$ we have $f(x)={h(x^{-1})\over h(x)}.$ Therefore $f(x)$ should be undefined at $x=1$ or at $x=-1,$ as otherwise $f(1)=f(-1)=1.$ Hence $h(x)$ ...
Ryszard Szwarc's user avatar
0 votes

evaluate: $ \lim_{n→∞} \frac1n ((n+1)(n+2)(n+3)⋯(2n))^{\frac1n}$.

This is a solution that a first year undergrad would be expected to give (no knowledge of Riemann integral or Stirling approximation yet beautiful in my opinion.) We want to evaluate: $$\ell :=\lim_{n\...
Nothing special's user avatar
1 vote

Prove that a function doesn't have a horizontal asymptote

hint Assume, it has. So, $$\exists a\in R :\lim_{x\to \infty} (f(x)-a)=0$$ $\implies$ $$\lim_{x\to \infty}(f(x+1)-f(x))=0$$ $\implies$ $$\lim_{x\to\infty} f'(C_x)=0$$ with MVT and $x<C_x<x+1$, ...
hamam_Abdallah's user avatar
0 votes

Domain $U$ for "if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative"

Let $U=U_1\sqcup U_2$ where $U_1,U_2$ are open simply connected. $F\vert_{U_i}$ is $C^1$. Then since $\nabla\times F\vert_{U_i}=0$, by Poincare's lemma, we have $\Phi_i$ on $U_i$ such that $\nabla \...
user1099762's user avatar
0 votes

Derivative rule for a function raised to the power of another function eg $(3x^2+5)^{\arctan{x}}$

Another solution $$ \begin{aligned} & y=f(x)^{g(x)} \\ & y^{\prime}=? \\ \text{Aplay natural log} \\ & \ln y=\ln f(x)^{g(x)}=g(x) \cdot \ln f(x) \\ \text{Then proceed to derive}\\ & \...
Time Step's user avatar
1 vote

Find the set of values of $\alpha$ so that $f(x)=\dfrac{\alpha x^2+6x-8}{\alpha+6x-8x^2}$ is one one.

Hint Notice that if $\pm 1$ are both in the domain of $f$, then $$f(\pm 1) = \frac{\alpha (\pm 1)^2 + 6(\pm 1) - 8}{\alpha + 6(\pm 1) - 8(\pm 1)^2} = \frac{\alpha - 8 \pm 6}{\alpha - 8 \pm 6} = 1 .$$ ...
Travis Willse's user avatar
2 votes
Accepted

Surface integral $\int \int_R \sqrt{1-x^2-y^2} \mathrm{d}x\mathrm{d}y$

Let $1-r^2=u$ so that $-2r\,\mathrm dr=\mathrm du$. Then, $$\int r\sqrt{1-r^2}\,\mathrm dr=\int-\frac12\sqrt{u}\,\mathrm du$$ Then, it is straightforward. Hope this helps. :)
ultralegend5385's user avatar
1 vote

How to simplify $\frac{\cos(x)+\sqrt{x}\sin (x)}{2 \sqrt{x} \cos^2(x)} $ into $\frac{\cos(x) +2 x \sin(x)}{2 \sqrt{x} \cos^2(x)}$?

These fractions are not equal, so you cannot simplify one into the other.
Grütter's user avatar
0 votes

Proving the sequence of the fractional part of the root of a natural number diverges

By contradiction, suppose there exists a limit $L$. One has $(\{\sqrt n\})_{n\in\Bbb N}\in [0,1)\Rightarrow L\in[0,1]$. By our assumption,for all $\epsilon\gt0$ there exists $N$ such that $n\gt N$ ...
Piquito's user avatar
  • 29.8k
1 vote

Find $\int_0^1\ln\left(\frac{1+x}{1-x}\right)~dx$ without resorting to series

Simply write the integral as $$\int_0^1\ln(1+x)\,\mathrm dx-\int_0^1\ln(1-x)\,\mathrm dx$$ Knowing that an antiderivative of $\ln x$ is $x\ln x-x$, the first integral becomes $$\left[(x+1)\ln(x+1)-(x+...
ultralegend5385's user avatar
2 votes
Accepted

Is this function differentiable? (Error in the notes?)

This function $f$ is only defined on $[0,\infty)\times\Bbb R$. If we define differentiability of $f$ at $(0,0)$ (which is only on the boundary) by mimicking the usual definition for points interior to ...
Anne Bauval's user avatar
  • 35.2k
1 vote

Prove $f$ attains its maximizer

For c), observe that $-f$ is coercive in the sense of b). In particular, the set $$C=\{x\in\mathbb{R}^n:-f(x)\leq-f(0)\}=\{x\in\mathbb{R}^n:f(x)\geq f(0)\}$$ is compact, and thus $f$ attains a maximum ...
Lorago's user avatar
  • 9,304
1 vote
Accepted

Prove $f$ attains its maximizer

As $\lim\limits_{||x||\to +\infty} f(x)=-\infty$ there exists some $M>0$ such that if $||x||>M$ then $f(x)<f(0)$. As $\overline{B(0,M)}$ is compact and $f$ is continuous, $f$ attains a ...
Julio Puerta's user avatar
  • 4,311
0 votes

Proving the sequence of the fractional part of the root of a natural number diverges

For $L=0$ pick $\varepsilon=\frac{1}{2}$ and $n=N^2-1$. Then it holds $$\varepsilon=\frac{1}{2}\le|a_n-L|=a_{N^2-1}=\sqrt{N^2-1} -(N-1),$$ since $$(N-1)+\frac{1}{2}<\sqrt{N^2-1}<N \ \...
user408858's user avatar
  • 2,462
1 vote

Proving the sequence of the fractional part of the root of a natural number diverges

We have that $\{\sqrt{n^2}\}=\{n\}=0$, so $a_{n^2}=0$ for every $n\in\mathbb{N}$. Thus, if the limit exists, it must be $0$. However, using the recommended bound we have $$n-\dfrac{1}{2}<\sqrt{n^2-...
Julio Puerta's user avatar
  • 4,311
7 votes
Accepted

Provide an ε-δ proof for the following limit

First notice that both $x^3-8$ and $x^2-4$ have a root at $x=2$. Moreover, we may express $x^3-8=(x-2)(x^2+2x+4)$ and $x^2-4=(x-2)(x+2)$. Therefore, if $x\neq 2$ and $x\neq -2$, we have $$\frac{x^3-8}{...
Julio Puerta's user avatar
  • 4,311
1 vote

A problem related to limit law

It is not applied here directly because it's of no use here. For example,you can't apply the limit provided to $e^n$ which is the numerator, as it individually tends to infinity. You need to first ...
Gwen's user avatar
  • 1,075
0 votes

functionally unfair coins

Essentially you are considering a sphere of diameter 1 and a coin taken from it as a calotte of height $h$, therefore of area $\pi h$ from Archimede theorem. The base of the calotte is a disk of ...
Letac Gérard's user avatar
6 votes
Accepted

Find the set of values of $\alpha$ so that $f(x)=\dfrac{\alpha x^2+6x-8}{\alpha+6x-8x^2}$ is one one.

Your strategy is good, and your computations are mostly correct, with a minor correction: $(1)$ implies that $$\color{red}{\alpha} \in [2, 14], \tag{1}$$ not $x$. And this is a correct conclusion, ...
heropup's user avatar
  • 136k
0 votes

Can this power series be solved using ratio test?

Ratio test: $$\lim_{n \to \infty}|\frac{(n+1)x^{n+1}}{2^{n+1}((n+1)^{2}+1)}\cdot\frac{2^{n}(n^{2}+1)}{nx^{n}}|$$ $$=\lim_{n \to \infty}\frac{(n+1)|x|}{2((n+1)^{2}+1)}\cdot\frac{(n^{2}+1)}{n}$$ $$=\...
bwootton's user avatar
  • 125
1 vote

$120\left(\int_{0}^{1} f\right)^{2} \leqslant \int_{0}^{1}\left(f^{\prime \prime}\right)^{2} $

Let $$I=\int_{0}^{1}f\mathrm{d}t$$ Since:$f(0)=f(1)=0, \quad$ Hence $$I=\int_{0}^{1}tf'\mathrm{d}t \quad \mathrm{and} \quad\int_{0}^{1}f'\mathrm{d}t=0 $$ Notice that:$$2I=\int_{0}^{1}t(1-t)f’’\mathrm{...
Yuechuan Rom's user avatar
0 votes

Circle area by integral definition

I wouldn't do it like that. It's much easier to calculate: $$\iint_\Omega dxdy$$ Where $\Omega = \{(x,y)\in\mathbb R^2 : x^2 + y^2 \le r^2\}$. Here you shall switch to polar coordinates. However, if ...
NtLake's user avatar
  • 150
1 vote

Which is bigger $\log_{2024}(2023)$ or $\log_{2023}(2022)$

Here's with differentiation: define $f(x):=\log_x(x-1).$ Then $f'(x)=(\frac {\ln (x-1)}{\ln x})'=\frac{\frac1{x-1}\ln x-\ln{(x-1)}\frac1x}{\ln^2x}=\frac{x\ln x-(x-1)\ln (x-1)}{x(x-1)\ln^2x},$ which ...
calc ll's user avatar
  • 8,511
1 vote

In layman's terms, what are curl and divergence?

You can visualize these concepts very well in fluid dynamics. Given a vector field $\mathbf F$, you can interpret it it as a velocity field of some fluid. Its divergence $\mathrm{div}(\mathbf F)$ at a ...
NtLake's user avatar
  • 150
2 votes

Which is bigger $\log_{2024}(2023)$ or $\log_{2023}(2022)$

Consider the function $f(x) = \frac{\log(x)}{\log(x+1)}$ for $x>1$. Then we have $$f'(x) = \frac{(x+1)\log(x+1)-x\log(x)}{x(x+1)\log^2(x+1)}$$ Then if you look at the function $g(t)=t\log(t)$, it ...
Tri's user avatar
  • 315
2 votes

Which is bigger $\log_{2024}(2023)$ or $\log_{2023}(2022)$

Let the difference $\log_{2024}{2023}-\log_{2023}{2022}$ be equal to $K$. On converting to base $e$, we get $$K=\frac{\ln(2023)}{\ln(2024)}-\frac{\ln(2022)}{\ln(2023)}$$ $$K=\ln\bigg(\frac{2023^2}{...
Sparsh Gupta's user avatar
0 votes

Calculating the integral: $\int \frac{dx}{\sin^3x}$

Yes, the simplified integral that you got is correct. Now, we are left with the integral $$I=-\int\frac{dv}{(1-v^2)^2}=\int\frac{-2vdv}{2v(1-v^2)^2}$$ Using integration by parts (IBP), $$I=\frac{-1}{...
Sparsh Gupta's user avatar
3 votes
Accepted

Integral inequality with specific condition

Note $$\begin{eqnarray} &&\int_0^1(f(x)+2-6x))^2 dx\\ &=&\int_0^1(f^2(x)+36x^2+4+4f(x)-12xf(x)-24x)dx\\ &=&\int_0^1f^2(x)dx-4\geq0 \end{eqnarray}$$ and hence $$ \int_0^1f^2(x)...
xpaul's user avatar
  • 44.1k
3 votes
Accepted

Integral inequality with exact values of a function

To relate $f(1)$ and $f(2)$ to the derivative $f'$ it is probably a good idea to use the fundamental theorem of calculus: $$\tag{1} f(2) - f(1) = \int_1^2 f'(x) \, dx.$$ This can be bounded above ...
Umberto P.'s user avatar
  • 52.3k
3 votes

Integral inequality with specific condition

Let $V$ be the subspace of $L^2[0,1]$ spanned by $\{1,x\}$. We are given that $\langle f,1 \rangle = \langle f,x \rangle = 1$. For simplicity, we will orthogonalize this basis by defining $e_1 = 1$, $...
whpowell96's user avatar
  • 5,595
2 votes
Accepted

why does estimation of this proper integral fail?

$\ln x$ pretends to be a power function, a very very weak power function. For example, $$ \int x^{n-1} \mathrm{d}x = \begin{cases} \frac1n x^n + C, & n \neq 0 \quad \text{(We are the family of ...
Y.D.X.'s user avatar
  • 364
3 votes

Help needed with an integral $\int_0^1 \frac{\ln(x) \ln(1+x) }{1+x^2} dx$.

Denote $I= \int_0^1 \frac{\ln x\ln(1+x)}{1+x^2}dx $, $J=\int_0^1 \frac{\ln^2(1+x)}{1+x^2}dx$ \begin{align} &\int_0^1 \frac{\ln^2(1-x)}{1+x^2}dx \overset{x\to\frac{1-x}{1+x}}= J -2 I-2G\ln2+\frac{\...
Quanto's user avatar
  • 97.6k
3 votes

Help request for calculating an integral

Substitute $x=\frac t{\sqrt{1+t^2}}$ \begin{align} \int \frac{2\sqrt{1- x^2}}{2 x\sqrt{1- x^2}+ 5}dx =& \int \frac{2}{(1+t^2)(5t^2+2t +5)}dt\\ =&\ \int \frac{t+\frac25}{t^2+\frac25 t +1}-\frac ...
Quanto's user avatar
  • 97.6k
2 votes

Help request for calculating an integral

Letting $x=\sin u $ we get: $$ \int \frac{2\cos^2 u}{2\sin u \cos u+5} du $$ Note that $2\cos^2 u =\cos(2u)+1$ and $2\sin u \cos u =\sin(2u)$ $$ \int {\cos(2u)+1\over\sin(2u)+5} du = \frac{1}{2}\int \...
Masd's user avatar
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