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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Using Jensen's Inequality

Let $f\in L^1(0,1)$. I want to show that $$ \left(\int_0^1 f(t) ~\text{d}t\right) ^2\leqslant \int_0^1f^2(t)~\text{d}t.$$ This is my attempt: I want to apply Jensen's inequality: $\varphi\left(\int_0^1 f\right) \leqslant \int_0^1…
Colin
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$f_n(x):= \frac{nx^3}{1+nx^2}$. Show that the sequence $(f_n)$ converges uniformly on $\mathbb{R}$

$f_n:\mathbb{R} \rightarrow \mathbb{R}$ A bit stuck on this revision question. I first determine the pointwise limit easy enough: $$\lim_{n\rightarrow \infty} \frac{nx^3}{1+nx^2} = x$$ To show that the sequence $(f_n)$ converges uniformly on…
JJJ
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How to conclude that the Taylor series of $f(x)=\log(1+x)$ is equal to $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} (n-1)!}{(1+x)^n}$

I want to find the Taylor series of $f(x)=\log(1+x), x \in (-1,+\infty), \xi=0, I=(-1,1)$ $$f'(x)=\frac{1}{1+x}$$ $$\sum_{n=0}^{\infty} (-1)^n x^n=\frac{1}{1+x}, x\in (-1,1)$$ $$f(x)=f(0)+\int_0^x f'(t) dt \Rightarrow f(x)=\int_0^x \frac{1}{1+x}dt=…
evinda
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How to find the Taylor series of $f(x)=\arctan x$.

I want to find the Taylor series of $f(x)=\arctan x,\; x\in[-1,1],\;\xi=0$. That's what I have tried do far: $$f'(x)=\frac{1}{1+x^2}=\frac{1}{1-(-x^2)}=\sum_{n=0}^{\infty} (-x^2)^n.$$ How can I continue?
evinda
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Derivative of $f(x)=\int_1^{\frac{1}{x}}\frac{\text{d}t}{\sqrt{(t^2-1)(1-t^2x^2)}}$

This function: $$f(x)=\int_1^{\frac{1}{x}}\frac{\text{d}t}{\sqrt{(t^2-1)(1-t^2x^2)}}$$ where $x\in (0,1)$ I want to find its derivative. In order to solve this question. I go back to general cases: $$f(x)=\int_a^{\varphi(x)}F(x,t)\text{d}t$$ By…
Laura
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Student question about compactness of sets

I'm just learning about compactness and being confused. Closed sets are not compact? Take $[a,b]$ and the covering $\{(a+\frac{b-a}n,b-\frac{b-a}n):n\geq3\}$. There is no finite subcover. $n$ must approach infinity otherwise it doesn't cover…
user144794
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properties of bounded function's second derivative

Is there a bounded function $f:\mathbb R\to\mathbb R$, such that $\forall_{t\in\mathbb R}$ $f''(t)\ne0$? If not, how to prove that?
user141766
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the series $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\sin(1+\frac{x}{n})}$ converges uniformly in $[-a,a]$

I have to show that the series $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\sin(1+\frac{x}{n})}$ converges uniformly in $[-a,a], a>0$. $$$$ That's what I have tried: $w_n=\frac{(-1)^n}{\sqrt{n}} \sin1…
evinda
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the series $ \sum_{n=1}^{\infty} \frac{1}{n^2+x^2}, x \in \mathbb{R}$ converges uniformly in $\mathbb{R}$ to a continuous $s(x)$

I am asked to show that the series $$ \sum_{n=1}^{\infty} \frac{1}{n^2+x^2}, x \in \mathbb{R}$$ converges uniformly in $\mathbb{R}$ to a $s(x)$ and that $s(x)$ is continuous. That's what I have tried: $$n^2+x^2 \geq n^2 \Rightarrow \frac{1}{n^2+x^2}…
evinda
  • 7,823
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Prove that $\frac{1}{n} \sum_{k=2}^n \frac{1}{\log k}$ converges to $0$

Prove that $\frac{1}{n} \sum_{k=2}^n \frac{1}{\log k}$ converges to $0.$ Okay, seriously, it's like this question is mocking me. I know it converges to $0$. I can feel it in my blood. I even proved it was Cauchy, but then realized that didn't…
badatmath
  • 4,065
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Prove that a set $E$ has content zero iff $E$ has a Jordan measure of $0$.

The following are the definitions I have been working with, A set $E \subset \mathbb{R}^n$ has $\textbf{content zero}$ if for every $\epsilon >0$ there exist $J_1,J_2, \dots, J_N$ "blocks" such that $E \subset \bigcup J_i$ and $\sum vol(J_i) <…
user7090
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Finding an upper bound smaller than 1

Let function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{1}{2}(x+\sqrt{x^2+1})$. I need to show that for each compact $K\subset\mathbb{R},\exists\alpha_K<1$ such that $|f(x)-f(y)|\leq\alpha_K|x-y|,\forall x,y \in K$. As $K$ is compact,…
user145306
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Constructing $\mathbb{R}$ from $\mathbb{Q}$ and showing $\mathbb{Q}$ is dense in $\mathbb{R}$

This is a very long, multi-part problem that we were told to figure out by any means possible. There are no limits on getting help or finding answers online. I haven't had much luck at all solving this for myself, can anyone help me with some of…
user137684
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Why is f bounded?

I am looking at the exercise: Let $f:[0,1] \to \mathbb{R}$ integrable such that $|f(x)| \leq \int_0^x |f(t)|dt, \forall x \in [0,1]$. Show that $f=0$. At the solution,it is taken that $f$ is bounded. Why??Because of the fact that $f$ is integrable?
evinda
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How can I prove this function is not continuous for every point other than 0?

Define $g:[0,1]\rightarrow\mathbb R$ by $g(x)=\sqrt{x}$ if $x$ is rational and $g(x)=0$ if x is irrational. Prove that $g$ is continuous at $x=0$, but is not continuous at any other value of $x$. I am really at a loss of where to even start my…
blubberbrot
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