Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
7
votes
1 answer

Show that there exists a $c : f(c) = g(c)$

I'll try to present a solution for this problem, and I hope I can receive feedback on what went wrong, if something went wrong of course. Let $f, g : [a, b] \to \Bbb R$ be continuous functions and $\int_{a}^{b} f(x) dx = \int_{a}^{b} g(x) dx$.…
Tanamas
  • 1,837
7
votes
2 answers

$\sum_{n\ge1}\frac{a_n}{(s_n)^\alpha} \text{converges} \iff \alpha > 1$ , where $\sum_{n\ge1} a_n$ is divergent

Let $\forall n \quad a_n >0, \quad s_n=\sum_{k=1}^n a_k$ If $\sum_{n\ge1} a_n$ diverges, than show that $$\sum_{n\ge1}\frac{a_n}{(s_n)^\alpha} \text{converges} \iff \alpha > 1$$
Yoshua Yonatan
  • 865
7
votes
1 answer

If $f:I\to\mathbb{R}$ is $1{-}1$ and continuous, then $f$ is strictly monotone on $I$.

Suppose that $I\subseteq\mathbb{R}$ is nonempty. If $f:I\to\mathbb{R}$ is $1{-}1$ and continuous, then $f$ is strictly monotone on $I$. The answer in the back of the book$^1$, which I found after writing the following proof, says this is false (no…
user80696
7
votes
6 answers

Constructive proof of boundedness of continuous functions

Consider the theorem for the continuous function: Let $a
user9464
7
votes
7 answers

Show that $n^2\log\left(1+\frac{1}{n}\right)$ does not converge to $1$

How to show that $n^2\log\left(1+\dfrac{1}{n}\right)\to 1$ is false? I have to show that $\left(1+\dfrac{1}{n}\right)^{n^2}$ doesn't tend to $e.$
Sriti Mallick
  • 6,137
  • 3
  • 30
  • 64
7
votes
1 answer

$(X,d)$ is metric space. $(X,d)$ is compact if and only if any continuous function on $X$ has a maximum.

$(X,d)$ is metric space. $(X,d)$ is compact if and only if any continuous function on $X$ has a maximum. I dont know whether these functions real valued or not but only real valued functions may make sense, I think. In that case $\Rightarrow$ is…
Jale'de jaled
  • 780
7
votes
1 answer

Is the space of real analytic functions a freely generated algebra?

Let us consider the space $C^{\omega}(\mathbb{R})$ of all the functions $f \colon \mathbb{R} \to \mathbb{R}$ which are analytic on the whole real line. It is clear that $\mathcal{C}^\omega(\mathbb{R})$ is an algebra, because it is closed under…
sebpar
  • 325
7
votes
3 answers

If $\lim\limits_{x \to \pm\infty}f(x)=0$, does it imply that $\lim\limits_{x \to \pm\infty}f'(x)=0$?

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is everywhere differentiable and $\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$, there exists $c \in \mathbb{R}$ such that $f(c) \gt 0$. Can we say anything about $\lim_{x \to \infty}f'(x)$ and…
user4167
  • 1,453
7
votes
4 answers

To show that function is constant

Let $f$ be defined on $\mathbb{R}$ and suppose that |$f(x)$ - $f(y)$| $\leq$ $(x-y)^2$ $x,y \in\mathbb{R}$. Here I have to show that $f$ is a constant function. I think I have to show that $f'(x)$ = 0 for all $x$. But I don't know from where to…
monalisa
  • 4,460
7
votes
1 answer

Does there exist a smooth function which is nowhere analytic?

Smooth means has derivatives of all order, and analytic means can be given as a convergence of power series.
van abel
  • 1,461
7
votes
4 answers

Limit of $n^{\frac{1}{n}}$.

We have to prove that $n^{\frac{1}{n}}$ converges to $1$. I have proved it using the binomial theorem where we can substitute $(1+t)$ in place of $n$ and proceed forward. However along with the question another approach was mentioned where we can…
V2002
  • 147
7
votes
3 answers

How many irrationals are there which are unique upto addition with rationals?

I'm studying real analysis and I know from previous courses that there are countably infinite rationals but uncountably infinite irrationals. However, I haven't done a formal proof about uncountability of irrationals. I've been thinking about which…
Peeyush Kushwaha
  • 707
7
votes
1 answer

the integral of $\frac{\sin x}{x}$

in the evaluation of $\int_{0}^{+\infty}\frac{\sin x}{x}$, there was such a strategy as to evaluate it as $\lim_{s \to \infty}\int_{0}^{s}\frac{\sin x}{x}= \lim_{s \to \infty}\int_{0}^{s}\sin x \int_{0}^{+\infty}e^{-xt}dt $, and then we change the…
Alex
  • 2,583
7
votes
2 answers

If $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent

Let $\sum_{n=1}^{\infty} a_n$ be a series in R. Prove that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent.
Lindsay Duran
  • 1,255
7
votes
4 answers

The proof of Raabe's Test for absolute convergence

In Introduction to Real Analysis second edition by Bartle & Sherbert's, there is a proof of Raabe's Test for absolute convergence. The problem is that I don't understand why some part of the proof is necessary. I will show you first the proof as it…
Spenser
  • 19,469
1 2 3
…
99 100