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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Mean Value Theorem related problem

Let $f(x)\leq g(x)$ for all $x \in I$, where $I$ is an interval $\subseteq$ R. Also, let $f(c) = g(c)$ for some $c \in I$ but not an endpoint. Prove that $f'(c) = g'(c)$ (assume differentiablity) I have tried the Mean Value Theorem, let I = [a,b].…
sucksatmath
  • 141
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Regarding $\lim_{n \to \infty} n^{\frac{1}{n}}$

Suppose $\lim_{n \to \infty} n^{\frac{1}{n}} = l \in \mathbb{R}$. The function $f(x) = x^n$ is continuous, then $$l^n=\left (\lim_{n \to \infty} n^{\frac{1}{n}} \right)^n=\lim_{n \to \infty} \left ( \left (n^{\frac{1}{n}} \right)^n \right ) =\lim_{n…
user4167
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$f(x)=x-x^2$ if $x$ is rational, $x+x^2$ if $x$ is irrational

$$f(x)= \left\{\begin{array}{ll} x-x^2 &\mbox{if $x$ is rational,}\\ x+x^2 &\mbox{if $x$ is irrational.} \end{array}\right.$$ Show that $f'(0)=1$ and yet there is no neighborhood $I$ of the point $0$ on which this function is monotonically…
Jihyun Kim
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Can any function of two variables be expressed as a linear combination?

Can any function of two variables be expressed as a linear combination (may be infinite) of a product of two single-variable functions? $f(x,y) = \sum A_n g_n(x) h_n(y)$
Some Student
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Continuous functions and open sets/covers

I'm taking a course in elementary algebraic geometry, but I seem to be lacking a topological background. The following result is often used: Let $X,Y$ be topological Spaces, and $U_i$ for $i \in I$ an open cover for $X$ (the convention here is that…
Albert
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Proof of a theorem with upper/lower limits.

Theorem: If $s_n \le t_n$ for all $n$ greater than a fixed integer $N$, then $$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n$$ I would like to prove this and it would be nice if someone could check my work. Proof: Letting…
hyg17
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I'm having trouble with a definition of the upper and lower limits, and a theorem that follows it.

The following is the definition. Let $\{s_n\}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ such that $s_{n_{k}}\rightarrow{x}$. This set $E$ contains all subsequential limits, plus possibly the numbers $+\infty$, $-\infty$.…
hyg17
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Finding the maximal $t$ satisfying a family of inequalities

Given $c \in (0,1)$, find the maximal positive $t$ satisfying the following: $$\forall n \in \{1,2,\ldots \}: 1+\frac{c}{n+(1-c)} \le \left(1+\frac{1}{n+t}\right)^{c}$$ My progress thus far: A special case is $c = \frac{1}{2}$: the inequality…
Ofir
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How to prove these

How to prove these (1) $\displaystyle\sum_{n=4}^\infty\frac{(-1)^n\ln n}n$ if it is absolutely or conditionally convergent? and (2) $\displaystyle\sum_{n=1}^\infty\frac n{(-2)^n}$ if it is absolutely or conditionally convergent? What I am thinking…
LoveMath
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Conjecture If f is surjective then there exists x $\in$ (a, b) such that $|f'(x)| = 1$

Conjecture Let $f$ be a continuous function from [a, b] to [a, b], and is differentiable on (a, b). If f is surjective then there exists x $\in$ (a, b) such that $|f'(x)| = 1$ Any counter example for this conjecture ? **Addition after Kavi Rama…
Pascal
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Probability of failure of a campaign

This is a simplistic question. It is about the issue of running a campaign in a newspaper. There are two issues here: (i) The chances of failure vs success of a well-planned, well-written, and well-excuted campaign page (in terms of leaving an…
Subir Ghosh
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looking for 8 digit numbers with 4 digits being used twice

I'm looking for $8$ digit numbers with $4$ digits being used twice. for example : $11223344$ and $12123434$ and $11002233$ Its not allowed to use one digit like: $11223345$ for $4$ digit numbers with $2$ digits being used twice I have computed $243$…
MasOOd.KamYab
  • 141
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Does the sequence converges?

I am trying to prove if the sequence $a_n=(\root n\of e-1)\cdot n$ is convergent. I know that the sequences $x_n=(1+1/n)^n$ and $y_n=(1+1/n)^{n+1}$ tends to the same limit which is $e$. Can anyone prove if the above sequence $a_n$ is convergent?…
LoveMath
  • 769
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2 answers

How to prove an alternating series is convergent?

How to prove that the sequence $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{\sqrt{n}}$ is convergent? I was trying to find the upper bound and lower bound of the partial sum $s_k$ and use Squeeze Theorem to figure out the limit, but I couldn't find the…
Lindsay Duran
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L'Hôpital's rule for vector-valued functions

What are the traps, when using L'Hôpital's rule in multivariable calculus to determine the lim of an vector function? I heard there some more cases, where L'Hôpital's rule in multivariable calculus is not applicable. Update: I meant multivariable…
Schwoisser
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