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Results tagged with real-analysis
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user 1684451
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.
4
votes
1
answer
155
views
Properties of continuous functions on $(0,\infty)$ with $f(x) = f(2x)$ for all $x$.
The problem:
Let $\mathscr{S}$ be the family of continuous real valued functions on $(0,\infty)$ defined by:
$$
\mathscr{S} :=
\left\lbrace f : (0,\infty)\to\mathbb{R} \,\middle|\, f(x) = f(2x), \fora …
- 342
1
vote
3
answers
184
views
Convergence of the improper integral $\int_{0}^{\infty} (\pi + x^{3})^{-1/4}\,\mathrm{d}x$.
Problem: Decide whether the improper integral
$$
\int_{0}^{\infty} (\pi + x^{3})^{-1/4}\,\mathrm{d}x$$
converges or diverges.
My attempt: I started by breaking the given integral into two as follows:
…
- 342
1
vote
0
answers
42
views
Book recommendations for rigorous multivariable calculus with emphasis on differential forms
I'm looking to study the analysis of differential and integral calculus for functions of several variables from a rigorous perspective, with a particular emphasis on differential forms.
My background …
- 342
1
vote
1
answer
57
views
A map defined on a compact domain is continuous if and only if its graph is compact. [duplicate]
Problem: If $E$ is a compact subset of a metric space $X$, and $f$ is a map defined on $E$ to a metric space $Y$, then prove that the graph of $f$, denoted by $G(f)$ and defined as the set
$$\{(x,f(x) …
- 342
4
votes
3
answers
206
views
$\lim n^{2}\left(\log{2} - \sum_{1\leq j\leq 2n-1}\frac{2}{2n + j}\right) = $?
Problem: Let
$$
a_{n} = \frac{2}{2n+1} + \frac{2}{2n+3} + \cdots \frac{2}{4n-1}.
$$
Then find the following limit:
$$
\lim_{n} n^{2}\left(\log{2} - a_{n}\right).
$$
My attempt: Note that
$$
\frac{2(2 …
- 342
2
votes
2
answers
104
views
Continuity of $\sum_{n = 1}^{\infty} e^{-nx}\sin{nx}$.
I was doing a problem where I was asked to show the continuity of
$$\sum_{n = 1}^{\infty} e^{-nx}\sin{nx},$$
for $x>0$.
My approach was to consider the sequence $(\sigma_{n}(x))$ of partial sums of th …
- 342
2
votes
2
answers
261
views
How to prove that the distance between a compact and a closed set that are disjoint is posit... [closed]
Problem: Suppose $K$ and $F$ are disjoint sets in a metric space $X$, $K$ is compact and $F$ is closed. Show that there exists a $\delta >0$ such that $d(x,y) > \delta$, for $x \in K$ and $y \in F$.
M …
- 342