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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Few questions (1.4, 1.5, 1.6) from Rudin 2e on roots of reals

I picked up a copy of Principles of Mathematical Analysis by Rudin. It just so happened to be the second edition which is drastically different from the third and I can't find solutions to a lot of exercises. I'd like some hints on how to complete…
stochasticats
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Intuitive explanation of why $n \sum_{i=1}^n (x_i - x)^2 = \sum_{i< j} (x_i - x_j)^2$, where $x = \sum_{i=1}^n \frac{x_i}{n}$

For real numbers $x_1, \dots, x_n$, we have the simple identity: $$ n \sum_{i=1}^n (x_i - x)^2 = \sum_{i< j} (x_i - x_j)^2$$ where $$x = \sum_{i=1}^n \frac{x_i}{n}$$ One can easily multiply out both sides to obtain the equality. Is there an easy…
JohnKnoxV
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Prove Cauchy sequence from

Can anyone prove from first principles, that $$a_n = \left(\frac{2n-1}{n}\right)_{n=1}^\infty$$ is a Cauchy sequence. Thank you all.
LoveMath
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euclids fourth postulate

Hi guys I just need to know if my answer is right. The question is 1) Euclids $4^{th}$ postulate is "That all right angles are equal to one another". Why is this not obvious? My answer: When I read this question I am like it is obvious, so I got…
MathGeek
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Connection between sets and functions

Let $A$ be a convex closed subset of $\mathbb R^n$. Is there a convex function $g: \mathbb R^n \rightarrow \mathbb R$ such that $ A=\{x: g(x) \leq 0\} ? $
user 531
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Prove the converges of the followin sequence and find the limit

I wonder if anyone can solve the following problem for me: The sequence of real numbers $x_n$ is defined inductively by $$ x_1=4 \quad\text{and}\quad x_{n+1}=\frac4{x_n}+\frac{x_n}2. $$ Show that $x_n$ converges and find its limit Thanks to every…
LoveMath
  • 769
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Find the limit to the following

Please if any one can find the following limit: Let $b$ be a real number, $b>1$. Compute $$ \lim_{n\to\infty}n\cdot\biggl(\frac1{b^n}+\frac1{b^{2n}}+\frac1{b^{3n}} +\dots+\frac1{b^{(n-1)n}}+\frac1{b^{n^2}}\biggr). $$
LoveMath
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Prove disprove that for $a$, $b$, $c$, $d$ are real numbers

Can any one solve the following two for me: 1- Let $a$, $b$, $c$ and $d$ be real numbers satisfying $a>b$ and $c>d$. Does this imply that $ac>bd$? Prove or disprove. 2- From the analysis concepts…
LoveMath
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Does there exist a continuous surjection $f:\mathbb{R} → l^{\infty}$

I have asked a question asking for a surjection from $\mathbb{R}$ to $\mathbb{R}^2$. Any thoughts, anyone? Or can someone start me off by offering me a hint how to find a surjection from $\mathbb{R}$ to $l^\infty$?
Lost1
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The proof of 'the limit of $\sin(1/x)$ as $x$ approaches $0$ does not exist'

How can I prove that the limit of $\sin(1/x)$ as $x$ approaches $0$ does not exist? Should I use $\varepsilon$-$\delta$ in order to prove it? Are there any alternative ways to prove it?
PaleBlueDot777
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Can all algebraic numbers be expressed as definite integrals of rational functions with rational coefficients over rational domains?

If this is possible, how would I go about proving this? It's ok for the function to be defined in multiple variables and use multiple integrals, given that the domain itself is also defined by rational functions.
intingfeeder
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Show the function is differentiable

$F(x) = \int_{x-1}^{x+1}f(t)dt$ for x an element of the reals. Show that $F$ is differentiable on Reals, and compute $F^\prime$. I am unsure about how to showing $F$ is differentiable. I know that I need to use the fundamental theorem of calculus,…
mary
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About interpretation of Landau notation.

I have a question about Landau notation for complex analysis. Let $f(z)$ be a complex function and $g(z)$ be a real valued function of complex valuable which satisfies $g(z) \ge 0$. Then we write $$f(z)=O(g(z))$$ when these functions…
yuu
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limit superior of function

Let $f : [2, \infty) \to [0,1)$ is a function with $$f(x) = \dfrac{\sqrt{4 + \left[\ln(1 + \sqrt{x^2 - 4})\right]^2} - 2}{x-2}$$ for all $x \in [2, \infty)$. Prove that, for all $x \in (2, \infty)$, the function satisfies $\limsup_{s \to x^+} f(s) <…
asrida
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Big-Oh question

Let $y \in \Bbb R_+$ be an arbitrary positive real number. Prove that if $f \in O(y)$, then $f \in O(1)$. My attempt: $f \leq c_1y$, then let $c_2 = c_1y$, therefore $f \leq c_2\times 1$. Is it correct? Thanks.
Shinyshinystar
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