Intereting Posts

About the identity $\sum\limits_{i=0}^{\infty}\binom{2i+j}{i}z^i=\frac{B_2(z)^j}{\sqrt{1-4z}}$
Difficult Integral: $\int\frac{x^n}{\sqrt{1+x^2}}dx$
Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.
Additive group of rationals has no minimal generating set
What is the area of the apollonian gaskets?
$g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.
Number of solutions of equation
Is there a subfield $F$ of $\Bbb R$ such that there is an embedding $F(x) \hookrightarrow F$?
Calculate the Wu class from the Stiefel-Whitney class
Approximating piecewise linear function
Visualization of 2-dimensional function spaces
Tail bound on the sum of independent (non-identical) geometric random variables
Exercise 3.39 of Fulton & Harris
Is this a valid deductive proof of $2^x \geq x^2$ for all $x \geq 4$?
Finding limit of a quotient

Let $g:[0,1]\mapsto\mathbb{R}$ be a continuous function, and $\lim_{x\to0^+}g(x)/x$ exists and is finite. Prove that $\forall f:[0,1]\mapsto\mathbb{R}$,

$$\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$$

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- Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$
- Existence of a sequence with prescribed limit and satisfying a certain inequality II
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- $\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show it that convergence and also find $\lim_{x \to \infty} \{a_n\}$
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- How to prove that $\lim \frac{1}{n} \sqrt{(n+1)(n+2)… 2n} = \frac{4}{e}$
- Convolution of compactly supported function with a locally integrable function is continuous?
- Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$
- A function is continuous if its graph is closed

I’m a little late to the game and see that this suggestion has been made, but let’s carry it out.

Let $y=x^n$, $x=y^{1/n}$, $dy = (1/n) y^{1/n} dy/y$. Then

$$n \int_0^1 dx \: f(x) g(x^n) = \int_0^1 \frac{dy}{y} y^{1/n} f(y^{1/n}) g(y)$$

As $n \rightarrow \infty$, $y^{1/n} \rightarrow 1 \forall y \in (0,1]$; that is, apart from $y=0$; however, since this is an isolated point (i.e., measure zero), then we have

$$\lim_{n \rightarrow \infty} n \int_0^1 dx \: f(x) g(x^n) = f(1) \int_0^1 dy \frac{g(y)}{y}$$

as was to be shown.

Let $f(x)$ be a function that possesses a power series expansion, such that $\forall f:[0,1]\mapsto\mathbb{R}$, then for $f(x) = \sum_{k \geq 0} a_{k} x^{k}$,

\begin{align}

L &= \lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx = \sum_{k=0}^{\infty} a_{k} \, \lim_{n \to \infty} n \, \int_{0}^{1} x^{k} \, g(x^{n}) \, dx.

\end{align}

Make the substitution $t = x^{n}$ to obtain

\begin{align}

L &= \sum_{k=0}^{\infty} a_{k} \, \lim_{n \to \infty} \int_{0}^{1} t^{\frac{k+1}{n} – 1} \, g(t) \, dt \\

&= \sum_{k=0}^{\infty} a_{k} \, \int_{0}^{1} \frac{g(t) \, dt}{t} \\

&= f(1) \, \int_{0}^{1} \frac{g(t) \, dt}{t}.

\end{align}

This is the desired result.

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- Looking to attain fluency in mathematics, not academic mastery
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