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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Existence of a measure

I need help in showing that if $\alpha$ and $\beta$ are measures defined on $\mathfrak{A}$, and $\beta \leqslant\alpha$ then there is a measure $\lambda$ on $\mathfrak{A}$ such that $\lambda=\alpha-\beta.$ Attempt: I started by defining…
jojo
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sum of infinite series with telescopes

I need help finding the sum of the infinite series $$\sum_{k=1}^\infty \frac{1}{n(n+1)(n+2)}$$ I have used the partial fraction decomposition to get this as the sum of $$\frac{-1}{k+1}+\frac{1}{2(k+2)}+\frac{1}{2k}$$ but don't know where to go from…
ak55
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How find this sum $\sum_{n=1}^{\infty}\frac{n}{[(2n)!]^2}$

Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{n}{[(2n)!]^2}$$ I think we can note $$\dfrac{n}{((2n)!)^2}=\dfrac{1}{2}\dfrac{2n}{((2n)!)^2)}=\dfrac{1}{2}\cdot\dfrac{1}{(2n)!\cdot(2n-1)!}$$
math110
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$\int_1^e \! \frac{\mathrm{d}x}{x\,\sqrt{-\ln \ln x}}$

$$\int\limits_1^e \! \frac{\mathrm{d}x}{x\,\sqrt{-\ln \ln x}}$$ I can't find any antiderivative, is it possible to calculate the definite integral?
Rummelluff
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Rate of convergence of $\left[ \left( \sum\limits_{i=j}^n {2i+1}\right)^{\frac{1}{2}}\right]$

I have $$a_{n} = \left[ \left( \sum\limits_{i=j}^n {2i+1}\right)^{\frac{1}{2}}\right]$$ for some fixed $j\geq1$, where the square brackets are the fractional part. Now I know $$\lim_{n \to \infty} a_{n} = 1$$ as for $n\geq…
user185885
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How to find the countably infinite intersection

$ E(n) $ is a subset of $\mathbb{R}$. If $E(n)$ is open when $n$ is even and closed when $n$ is odd, and $E(n+1) \subseteq E(n)$ then $\cap_{n=1}^{\infty}E(n)=?$ Tried: It's essentially the same as asking $\lim_{n\to\infty} E(n)=?$ Let $E(1) =…
MatheMagic
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Find a compression of the frames over the subspaces

Find a compression of the frames $B=\{(1,1,0),(1,-1,0),(1,1,1),(0,0,1),(0,1,-1)\}$ over the subspaces (a) $M=\{(x,y,z):x=0\}$, (b) $M=\{(x,y,z):x=y\}$ Find an orthonormal basis $B$ of $\mathbb{R}^4$ such that the frame…
Username Unknown
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Is my method finding $\sup A$ and $\inf A$ fully correct?

$A=\left\{\dfrac{1}{n}+\dfrac{1}{n^2} \mathrel{\bigg|} n\in \mathbb N^*\right\}$ I have derived the function and I found $\dfrac{-n(n+2)}{n^4}$, so the function is strictly decreasing. Then I simply said: to find the maximum value for this function…
Tébina1
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Cantor-Lebesgue Functions

Show that there is a continuous, strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero. Is it enough to prove that a strictly increasing function that is defined on an interval has a…
J.R.
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Connections between two definitions of locally Lipschitz mapping

Let's consider two following definitions of locally Lipschitz mapping. Let $f: D\subset \mathbb R^n \rightarrow \mathbb R$. We say that $f$ is localy Lipschitz in 1. sense, if for each $a\in D$ there exist a neighbourhood $U_x$ of $a$ and a…
user172903
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How find this sum $I\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)}=1-\frac{1}{2}\ln{(2\pi)}$

Question: show that $$I=\sum_{k=1}^{\infty}\dfrac{B_{2k}}{2k(2k-1)}=1-\dfrac{1}{2}\ln{(2\pi)}$$ where $B_{n}$ is Bernoulli number:Bernoulli number I think we can $$I=\sum_{k=1}^{\infty}\left(\dfrac{1}{2k-1}-\dfrac{1}{2k}\right)B_{2k}$$ then I…
math110
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Continuity of integral

How can I prove the continuity or discontinuity of the following $\mu$ as a function of $x$? $$\mu(x) = \int_0^x\exp\left\{ -\int_0^t\frac{d\alpha(s)}{\beta(s)+1} \right\}dt + \int_x^{+\infty}\exp\left\{ -\left(\int_0^x\frac{d\alpha(s)}{\beta(s)+1}…
pelusos
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Is locally Lipshitz mapping on a compact set globally Lipschitz?

Let $f: K \rightarrow \mathbb R$ be a locally Lipschitz mapping on a compact subset $K$ of $\mathbb R^n$. Is it then $f$ Lipschitz?
user172903
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Given two sets $A,\; B$ and that $|A| = |B|$, show that $|2^A| = |2^B|$

Given two sets $A,\; B$ and that $|A| = |B|$, show that $|2^A| = |2^B|$. Intuitively, I think this is true, but I am having trouble showing this formally. I know that there exists a bijection $f: A \to B$ and that we should try to define a map $g:…
nomly
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Taylor theorem for function of several variables

Where one could find the proof of the following version of Taylor theorem for functions of several variables? Assume that $f$ is a function of class $C^{n+k}$ defined in a neighbourhood $W$ of zero in $\mathbb{R}^m$ with values in $\mathbb{R}.$ …
A.B
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