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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Prove that if $s_n ≤ b$ for all but finitely many $n$, then $\lim s_n ≤ b$.

The question asks me to prove that if $s_n ≤ b$ for all but finitely many $n$, then $\lim s_n ≤ b$ where $(s_n)$ be a sequence that converges. . Here is how I did it but im not sure if its entirely correct. I used proof by contradiction. Suppose…
user60887
  • 2,935
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Is integration continuity-preserving?

I am curious about the following question: Suppose that $f(x;\theta)$ is a bounded function of $x$, where the domain of $x$ is $[a,b]\subseteq \mathbb{R}$, and $\theta$ is viewed as a parameter. If $f(x;\theta)$ is continuous in $\theta$, then under…
OnoL
  • 2,685
4
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2 answers

Can a function be uniformly continuous on an open interval?

I am learning analysis and all the uniformly continuous functions I have seen are over a closed interval. So, can a uniformly continuous function be defined on an open interval?
Jazz
  • 41
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Question regarding $\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right)$

I wanted to find out whether the following limit exists, and find the value if it does. $$\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right).$$ Attempt After many attempt to prove that the limit exists, I looked up…
user4167
  • 1,453
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1 answer

Closed form of an integral

Is there a closed form of $$\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx \quad \text{?}$$ I just know that $\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx = 0.514042...$
Sebamed
  • 53
4
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3 answers

power series radius of convergence

How can I prove that if the coefficients $\{a_k\}$ of the power series $\sum_{0}^{\infty} \{a_k\}x^k$ form a bounded sequence, then the radius of convergence is at least 1?
ninja
  • 73
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Counterexamples in Double Integral

I need to: $a.$ Give an example of function $f:\mathbb{R\times R}$ $\to$ $\mathbb{R}$ with domain in $[0,1]^2$ so that double integral exists but the function is not Riemann integrable. $b.$ Give an example (if any) for a non-integrable function…
user77440
  • 73
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Question on Showing points of discontinuities of a function are removable (or not)

The question is as follows: Given function: $F(x,y)=\frac{x + 2y}{sin(x+y) - cos(x-y)}$ Tasks: a/ Find points of discontinuities b/ Decide if the points (of discontinuities) from part a are removable Here is my work so far: (1) For part a, I…
Cecile
  • 888
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1 answer

Can anyone show or clarify

Can any anyone clarify or prove that if the derivative of a function $f$ is strictly positive then the function $f$ is strictly monotone increasing. I am really sure that the converse is not true as the converse will not be true for the function…
LoveMath
  • 769
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2 answers

Proof using Rolle's Theorem to show there is c such that f$^4$(c) = 0, for a < c < b

The question is as follows: Give 3 information: (1) f is a polynomial (thus I claim f is continuous at every point) (2) $f(a) = f'(a) = f''(a) = f'''(a) = 0$ (3) $f(b) = 0$ Goal: use Rolle's Theorem to show that there is c satisfying $a < c <…
Cecile
  • 888
4
votes
1 answer

If two normed spaces are Lipschitz equivalent, then one if complete iff the other is

Prove that if two normed spaces are Lipschitz equivalent, then one if complete iff the other is. My thoughts: Let $ (V_1, \Vert\cdot\Vert_1) $ and $ (V_2, \Vert\cdot\Vert_2) $ be Lipschitz equivalent normed vector spaces. Then there exists $f : V_1…
user938272
  • 923
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Inequalty with complex numbers

i'm being fighting with this for a long, it shouldn't be that hard, can you provide some ideas? Let a,b on $\mathbb{C}$ if |a|<1 and |b|<1 prove: $$\left|\frac{a-b}{1-\bar{a}b}\right| < 1$$ Thanks!
aramirezreyes
  • 417
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3 answers

Why are these two definite integrals equal?

How can one prove that, for $0< z<1$, the two integrals $$\int_0^\infty \frac{u^{z-1}}{1+u}du$$ and $$\int_0^\infty \frac{u^{-z}}{1+u} du$$ are equal? From the integral representation of the beta function $$B(z,w)=\frac12\int_{0}^\infty…
Nasser Saad
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non measurable sets

In Royden's text Vitali's theorem states that "Any set E of real numbers with positive outer measure contains a subset that fails to be measurable". So my question the Cantor set is a set of real numbers with outer measure zero does this set have a…
CLARE
  • 43
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Is my argumentation for proving that a monotonic sequence is convergent if and only if it is bounded enough?

I am studying for my first exam in Analysis 1 and one I have to know how to prove that a monotonic sequence in $\mathbb{R}$ is convergent if and only if it is bounded. I start by proving the first implication, thus that monotonic and convergent =>…
Mathias
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