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 .

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Infinite Sum of $(e^{ir} \arctan(r \sqrt(\pi)))/r^2$

I'm trying to evaluate this sum. $$S = \sum^\infty_{r=1} \left(\left(\frac{e^{ir}}{r^2}\right)\left(\tan^{-1}(r\sqrt{\pi})\right)\right)$$ I used the exponential definitions of $\sin(\theta)$ and $\cos(\theta)$ to get it into the form... $$S =…
Paarth S.
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How can I find the example of $f(x)$ such that $\,\lim_{x\to\infty}f(x) \neq 0$?

Let $f:[0,+\infty)\longrightarrow R^{+}\bigcup\{0\}$ be a continous and for any $x\in[0,+\infty)$ the sequence $\{f(x+n)\}$ converges to zero,prove that $$\lim_{x\to+\infty}f(x)=0$$ I think this problem is wrong, so someone can take some…
math110
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Verify some ideas from Munkres Analysis on manifold question 13.7

This is the problem from Munkres's Analysis of manifold 13.7. I'm not sure about this problem so I want to verify my ideas about the first 3 parts of this question. Then I hope there can be some help with the last part. For the notation: $$f_S(x)=…
M_k
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Is $x \in l_p$ for every $p$?

If $x$ is sequence such that $x_k=\sin(\frac{1}{\sqrt[3]k})-\sin(\frac{1}{\sqrt[3](k+1)})$ are coordinats, prove that $x \in l_p$ for every $p \ge1$ So, I have to prove $\sum_k (\sin(\frac{1}{\sqrt[3]k})-\sin(\frac{1}{\sqrt[3](k+1)}))^p$ is…
stranger
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Exercise about convergence/divergence of a series $\sum_{n=1}^\infty \frac{1}{n^2\sin n}$

I am asking if this series $$ \sum_{n=1}^\infty \frac{1}{n^2\sin n} $$ is convergent or divergent? To "study" this I used the $n$th term test wich says: if the limit of $ \frac{1}{n^2\sin n}$ isn't zero or it doesn't exist, than the series has to be…
user1103524
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Searching for a Berestycki reference

I would like to know if anyone has the following reference "METHODES TOPOLOGIQUES ET PROBLEMES AUX LIMITES NON LINEAIRES" published by Henri Berestycki in 1975. Thank you in advance.
lmf_math
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How is glb property implied from lub for set of all numbers of the format 1/n?

Consider the set $$ S = { \frac{1}{x} : x \in \mathbb{N} } $$ This set has the Least upper bound property since every non empty subset has a least upper bound in S. How is the greatest lower bound property implied in this case? Consider the subset…
urmish
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For $(X, d)$ and $f \colon X \to \mathbb R$, $\sup\,\{f(x) \mid x \in X\}-\inf\,\{f(x) \mid x \in X\}=\sup\,\{|f(x)-f(y)| \mid x, y \in X\}$

I would like to show that for a metric space $(X, d)$ and a function $f \colon X \to \mathbb R$, $\sup\,\{f(x) \mid x \in X\}-\inf\,\{f(x) \mid x \in X\}=\sup\,\{|f(x)-f(y)| \mid x, y \in X\}$. So far I have: Since for all $x, y \in X$,…
user83343
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Convergence of general improper integral

Let $h$ -- positive, twice continuously differentiable function on the $x\ge0$, such that $h'(x)>0$ and $h''(x)=o((h'(x))^2),$ $x\to\infty$. Show that $\displaystyle\int\limits_0^\infty e^{-h(t)}dt$ converges. My attempt $$\int\limits_{x_0}^A…
thing
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Let X be a metric space. suppose cl(Z)=X. Show that if $Y\subseteq X$ is nonempty and open, then $Z\cap Y\neq \emptyset$

I am now taking a theory course in microeconomics and my professor gave us some problems on analysis. I am having a hard time getting used to it. This is my proof Suppose $$ Z\cap Y=\emptyset \implies Z\subseteq X\setminus Y$$ take $cl()$ on both…
slowpoke
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Theorem 3-8, Spivak Calculus on Manifolds

This is a question about the proof of Theorem 3-8. Theorem 3-8: Let $A$ be a closed rectangle and $f:A\rightarrow \mathbb R$ a bounded function. Let $B = \{x:f\hspace{1mm} \text{is not continuous at}\hspace{1mm} x\}$. Then $f$ is integrable if and…
Rmal
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Convergence of a series knowing information of convergence of the summands

I have the series $F(n)=\sum_{j=0}^n = f(n-j)g(j)$ and I have that 1- As $m$ increases, $f(m)$ approaches $L$ (the limit point) 2- As $j$ increases, $g(j)$ decreases EXPONENTIALLY to $0$ Question: Can I say that $\lim_{n\to \infty} F(n) =…
user87757
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Implicit Function Theorem, proof from Spivak's Calculus on Manifolds

On page 42, Spivak proves the Implicit Function Theorem. As shown in the picture, $k$ is a function defined on $W$, but how I can know that $\forall x\in A, (x,0)\in W$? This is the only part of the theorem that I do not understand.
Rmal
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How can i prove this question without using The Mean Value Theorem?

Question: Suppose $f'$ is continuous on $[a,b]$ and $\epsilon\gt0$.Prove that there exist $\delta\gt0$ such that $|\frac{f(t)-f(x)}{t-x}-f'(x)|\lt\epsilon$ whenever $0\lt|t-x|\lt\delta$,with $x$ and $t$ in $[a,b]$. My question is how can I prove it…
G.t.g.h
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Let $A\subset\mathbb{R}$ be an uncountable set. Prove that $A\cap A'\neq\emptyset$.

$A'$ is the set of accumulation points of $A$. I first proved that $A'\neq\emptyset$, after I tried construct a Cauchy's sequence to show anything, but it's not so useful...
Kempa
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