Intereting Posts

What happens to small squares in Riemann mapping?
Determining k: $\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x – 16}} = \frac{\pi}{k}$
Induction proof. Explain in detail why it’s incorrect
How can there be genuine models of set theory?
Stuck with proof for $\forall A\forall B(\mathcal{P}(A)\cup\mathcal{P}(B)=\mathcal{P}(A\cup B)\rightarrow A\subseteq B \vee B\subseteq A)$
Does the limit of $e^{-1/z^4}$ as $z\to 0$ exist?
Dr Math and his family question. How to solve without trial and error?
Why is every meromorphic function on $\hat{\mathbb{C}}$ a rational function?
The set that only contains itself
Completion of rational numbers via Cauchy sequences
The Hexagonal Property of Pascal's Triangle
Does this hold: $(1+\sqrt{M})^n=2^n\cdot\sqrt M$
$A$ is skew hermitian, prove $(I-A)^{-1} (I+A)$ is unitary
Choice of $q$ in Baby Rudin's Example 1.1
What is the value of $\aleph_1^{\aleph_0}$?

How do i prove the limit below? I’ve tried, but i got nothing.

$ \lim\limits_{n\to \infty} (3^n + 4^n)^{1/n} = 4. $

Thanks for any help.

- What does $dx$ mean?
- Prove the general arithmetic-geometric mean inequality
- Length of a Parabolic Curve
- Compute: $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)\cdot (2n+1)}$
- Closed-form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$
- Difficult Gaussian Integral Involving Two Trig Functions in the Exponent: Any Help?

- Prove that all values satisfy this expression
- Proving that $\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos (\tan (x))-\cot (x))\cot (x) \, dx=\frac{\pi(e-2)}{2e}$
- Show that every monotonic increasing and bounded sequence is Cauchy.
- Convergence or Divergence using Limits
- integration as limit of a sum
- Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $
- Mean value theorem application for multivariable functions
- Possible generalizations of $\sum_{k=1}^n \cos k$ being bounded
- If $a_1,\ldots,a_n>0$ and $a_1+\cdots+a_n<\frac{1}{2}$, then $(1+a_1)\cdots(1+a_n)<2$.
- How to calculate limit without L'Hopital

In general, let $\alpha_1,\alpha_2,\dots,\alpha_m$ be positive numbers. Let $A=\max\limits_{1\leq i\leq m}{\alpha_i}$

Then $$A^n\leq \alpha_1^n+\cdots+\alpha_m^n\leq mA^n$$

Thus $$A\leq (\alpha_1^n+\cdots+\alpha_m^n)^{1/n}\leq m^{1/n} A$$

So $$\lim_{n\to\infty} (\alpha_1^n+\cdots+\alpha_m^n)^{1/n}=\max\limits_{1\leq i\leq m}{\alpha_i}$$

since $m^{1/n}\to 1$.

Hint 2: $4^n < 3^n + 4^n < 2\cdot 4^n$

How about:

$$=\lim_{n \to \infty} 4\left(1+\frac{3^n}{4^n}\right)^{1/n} = \lim_{n \to \infty} 4\left(1+0.75^n\right)^{1/n} = 4$$

Hint: as $n \to \infty$, $4^n$ becomes much larger than $3^n$.

$$(3^n+4^n)^{1/n}= e^\frac{\ln(3^n+4^n)}{n}= e^\frac{\ln(4^n(1+(3/4)^n)}{n}= e^{\ln(4)}e^{\frac{1}{n}\ln(1+(3/4)^n)}\longrightarrow_{n\to \infty} 4 $$

- How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$?
- How to derive the law of cosines without the pythagorean theorem
- Infinite series expansion of $\sin (x)$
- How to evaluate Ahmed's integral?
- Calculating norm of a $2 \times 3$ matrix
- invariance of integrals for homotopy equivalent spaces
- Is it true that all single variable integral that had closed form can solve by one algorithms?
- $\mathbb{R}^3$ \ $\mathbb{Q}^3$ is union of disjoint lines. The lines are not in an axis diretion.
- The constant distribution
- Pigeonhole principle application
- Average minimum distance between $n$ points generate i.i.d. uniformly in the ball
- Trigonometric Inequality. $\sin{1}+\sin{2}+\ldots+\sin{n} <2$ .
- Confusing about coordinate curves and quadrilateral formed?
- Surface of genus $g$ does not retract to circle (Hatcher exercise)
- Interpretations of the first cohomology group