Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Showing that $\sum \frac{\log n}{n^x}$ converges for $x>1$

I'm trying to show that $\sum \frac{\log n}{n^x}$ converges for $x>1$ by the ratio test. Here's what I've got so far $$\frac{a_{n+1}}{a_n} = \frac{\log (n+1) n^x}{(n+1)^x \log n}$$ $$=\left(\frac{n}{n+1}\right)^x \frac{\log (n+1)}{\log n}$$ but I…
user26069
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Analysis Problem: Prove $f$ is bounded on $I$

Let $I=[a,b]$ and let $f:I\to {\mathbb R}$ be a (not necessarily continuous) function with the property that for every $x∈I$, the function $f$ is bounded on a neighborhood $V_{d_x}(x)$ of $x$. Prove that $f$ is bounded on $I$. Thus far I have…
Quaternary
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Proof of uniform continuity

I seem to have hit a dead-end in the following proof. Define $f:\mathbb{R}\to\mathbb{R}$ by: $f(x)=\frac{1}{1+x^2}$ Show that $f$ is uniformly continuous. My proof: Let $x_{0}\in \mathbb{R}$. Also let $\epsilon >0$ Choose $\delta = ?$ Then, for…
dplanet
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uniform approximation by smooth functions

Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u \in C^{k}(M)$. Can I always find a sequence of $C^\infty$ functions $\{u_n\}$ such that $u_n$ converges to $u$ in $C^k$ norm?
Q-the-curvature
  • 517
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Terence Tao, Analysis 1. Exercise 5.3.2. Real Numbers and Cauchy Sequences.

Let $ x = \lim_{n\rightarrow\infty}a_n, y = \lim_{n\rightarrow\infty}b_n$, and $ x' = \lim_{n\rightarrow\infty}a'_n$ be real numbers. Then $xy$ is also a real number. Furthermore, is $x=x'$, then $xy = x'y$. Here is my attempt. We need to show that…
user197848
5
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How prove there exsit $\xi\in (0,1)$ such $|f(\xi)|\le|f'(\xi)|$

Let $f:[0,1]\to \mathbb{R}$ be a differentiable function such that $f(1)=0$, Prove that there is $\xi\in(0,1)$, such that $$|f(\xi)|\le|f'(\xi)|.$$ My idea: I think we can prove there exsit $\xi\in (0,1)$ such $$(f(\xi)-f'(\xi))(f(\xi)+f'(\xi))\le…
math110
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How prove this limits is exsit $\displaystyle\lim_{n\to\infty}x_{n}$

let $f:[a,b]\to [a,b]$ be Continuous function,Assmue that sequence $\{x_{n}\}(n\ge 0)$ such $$x_{0}=x,x_{1}=f(x_{0}),x_{2}=f(x_{1}),\cdots,x_{n+1}=f(x_{n}),\forall n\in N^{+}$$ and $$\lim_{n\to\infty}(x_{n+1}-x_{n})=0$$ show that: …
math110
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5
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Can anyone clarify the meaning of zero content?

I am having hard time understanding the definition of zero content. The following are the definitions of zero content in $\mathbb{R}$ and $\mathbb{R}^2.$ A set $Z \subset \mathbb{R}$ is said to have zero content if $\forall \epsilon > 0$ there is a…
eChung00
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How prove there exist $\xi$ such $3af'(a)=3f(a)+a^3f'''(\xi)$

let $f(x)$ have three derivatives on $[0,a]$,and such $f(0)=f''(0)=0$, show that: there exsit $\xi\in[0,a]$ such $$3af'(a)=3f(a)+a^3f'''(\xi)$$ I think we can use Tarlor to solve it, $$f(0)=f(a)+f'(a)(-a)+f''(a)/2\cdot…
math110
  • 93,304
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3 answers

How find this value$\sum\limits_{n=0}^{\infty}\frac{(2n)!}{(n!)^22^{3n+1}}$

show that: $$\sum_{n=0}^{\infty}\dfrac{(2n)!}{(n!)^22^{3n+1}}=\left(\frac{1}{2}\right)^{1/2}?$$ this sum is from other problem,if I solve this,then the other problem is solve it
math110
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2 answers

Bijection Between $\mathbb{Z}_+$ and a Subset of $\mathbb{Z}_+ \times \mathbb{Z}_+$

Let $T=\{(a,b)\mid a,b\in\mathbb Z_+, b\leq a\}.$ Find a bijective function $f: T \to \mathbb{Z}_+$ I have tried to find a function but I can't, how does such function look like?
Matt
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What is a solution of such equation concerning the arithmetic and integral means?

Let $f:[a,b] \rightarrow \mathbb R$ be integrable and satisfies $$ f\left(\frac{x+y}{2}\right)=\frac{1}{y-x} \int_x^y f(t)dt $$ for all $x \neq y$, $x,y \in [a,b]$. What about $f$? Is it affine function? Thanks
Richard
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how to prove that $k+1 \ge (1+\frac{1}{k})^{k}$?

How to prove that $$k+1\ge \bigg(1+\frac{1}{k}\bigg)^{k} $$ when $k>2$
Sona
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How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$

Let $f(x)$ be a twice-differentiable function on $(a,b)$,show that there exsit $\xi\in(a,b)$ ,such $$\int_{a}^{b}f(x)dx=\dfrac{1}{2}(b-a)[f(a)+f(b)]-\dfrac{1}{12}(b-a)^3f''(\xi)$$ if this problem condition is Amuss that $f(x)$ is a…
math110
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4
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1 answer

Fourier transform of a compactly supported function.

Can someone help me the question below? Is there a positive-valued compactly supported function $f$ such that the Fourier transform ${{f}^{\operatorname{ft}}}\left( t \right)=\int_{-\infty }^{\infty }{f\left( x \right){{e}^{itx}}dx},…
Cao
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