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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Does the improper integral $\int_0^1 \frac{1}{x}\sin\frac{1}{x^2}$ exist?

Does the improper integral $\displaystyle\int_0^1 \frac{1}{x}\sin\frac{1}{x^2}$ exist? I don't know how to use the comparison test, and I cannot find a proper comparison function.
Knt
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Showing divergence of the series.

I am having hard time trying to show that the following series is divergent. Can someone help me please? $$\sum \frac{(-1)^n}{\log n} b_n $$ where $b_n=\frac{1}{\log n} $ if n is even and $b_n=\frac{1}{2^n}$ if n is odd. This would be a great…
user67035
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Evaluate $\lim_{x\to 0+}\sum_{n=2020}^{\infty}\frac{(-1)^n}{\log^x(n)} $.

Evaluate $\displaystyle\lim_{x\to 0+}\sum_{n=2020}^{\infty}\frac{(-1)^n}{\log^x(n)}$. It's easy to check that $$ \lim_{x\to 0+}\log^x(n)=1. $$ And for fixed $x$, since this is a decreasing alternating series, it converges obviously. Then I don't…
Knt
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$f(1)=\frac{1}{2}\int_{0}^{\frac{1}{2}}e^{1-x^2}f(x)dx$, prove that there exists $\xi\in(0,\,1)$ such that $f'(\xi)=2\xi f(\xi)$.

$f(x)$ is continuous on $[0,\,1]$, differentiable on $(0,\,1)$, and satisfies $$ f(1)=\frac{1}{2}\int_{0}^{\frac{1}{2}}e^{1-x^2}f(x)dx. $$ Prove that there exists $\xi\in(0,\,1)$ such that $f'(\xi)=2\xi f(\xi)$. Let $F(x)=e^{1-x^2}f(x)$, then…
Knt
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Computing accumulation points of a sequence in $\mathbb{R}^n$ programatically

I am dealing with a problem where the optimal solution of the problem is any accumulation point of the sequence the algorithm generates. Can someone point me to algorithms that are used to determine accumulation points of a sequence in…
batwing
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Give an example of a function $f$ that is differentiable at $x=a$ such that $f'(a)\ne0$ but yet $f$ attains a relative extremum at $x=a$.

Give an example of a function $f$ that is differentiable at $x=a$ such that $f'(a)\ne0$ but yet $f$ attains a relative extremum at $x=a$. By definition, the best I've gotten so far is using a function like $f(x)=x$ for $x=[0,1]$ since the maximum on…
user65384
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Ross' Elementary Analysis - 11.7

Let $(r_{n})$ be an enumeration of the set $\mathbb{Q}$ of all rational numbers. Show that there exists a subsequence $(r_{n_{k}})$ such that $\lim_{k \to \infty} = +\infty$. Theorem If the sequence $(s_{n})$ is unbounded above, it has a…
skm
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prove $\left(\frac{1}{\arctan x}-\frac1x\right)\lt x$ for $x\gt 0$

How to prove $\left(\frac{1}{\arctan x}-\frac1x\right)\lt x$ for $x\gt 0$ without the method of using its derivative? It appeared as a part of a math paper. The author said it could be easily proven by its Taylor's series. But I couldn't figure it…
Orion Pax
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Is the collection measurable sets of zero measure is a sigma algebra?

Given a measure space $(X,\Sigma,\mu)$, let us define the collection $\mathscr{C} = \{S \in \Sigma \:{:}\: \mu(S)=0\}$. I would like to show that $\mathscr{C}$ is a $\sigma$-algebra. I have managed to show that $\mathscr{C}$ is closed under the…
James
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Real roots for analytic function

Is there a generalized criteria for checking the reality of (all) roots of analytic function ? There are several such existing result for special case of polynomials like Newton's identities or Hermite Sylvester theorem or insights gained from…
user666369
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Representing 3D Surfaces as Music

Is it possible to represent 3D surfaces the same way we can represent sound, so that analysing sound is equal to analyzing a specific 3D model? For example, music is often decomposed into frequencies via Fourier Analysis. These frequencies can then…
Theodor Johnson
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$\lim\limits_{r\rightarrow \infty , 0} \left(\frac{\sum_1^n i^r/ n}{\sum_1^{n+1} i^r/(n+1)}\right)^{1/r} $?

Edited please help me! how can I evaluate: $$\lim_{r\rightarrow 0} \left(\frac{\sum_1^n i^r/ n}{\sum_1^{n+1} i^r/(n+1)}\right)^{1/r}$$ and $$\lim_{r\rightarrow \infty} \left(\frac{\sum_1^n i^r/ n}{\sum_1^{n+1} i^r/(n+1)}\right)^{1/r}$$
Math 1988
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Rudin Analysis: order in set of rationals

In Rudin's Analysis, chapter 1, page 3, definition 1.6 states $\mathbb Q$ is an ordered set if $r
Shiladitya Mukherjee
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Attempt at Proving A Lemma (critical point, 2nd derivative, global maximum).

could you please check my attempt at proving the following lemma. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ has two continuous derivatives, has only one critical point $x_{0}$ and $f''(x_{0})< 0$. Then $f$ achieves its global maximum at…
user4167
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Characterising continuity of a Riemann integrable function in terms of the upper and lower sums

Given a bounded Riemann integrable function $f:[a,b] \to \mathbb{R}$, I would like to know if there is any way to characterise the continuity of $f$ in terms of the difference between the upper Riemann sum $U(f,P_n)$ and the lower Riemann sum…
James
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