# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

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### A converse of sorts to the intermediate value theorem, with an additional property

I need to solve the following problem: Suppose $f$ has the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists a value $d$ between $a$ and $b$ for which $f(d)=c$, and also ...
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### Decimal representation series

Given the following sequence $d(n) = x \cdot n - \lfloor x \cdot n \rfloor$ with $n \in \mathbb{N}$, $x \in \mathbb{R} \backslash \mathbb{Q}$ . I want to show that for every $z \in [0,1)$ there is a ...
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### When is an infinite sum of polynomials analytic?

Consider the following function: $f(x) = \sum_{i,j>=0} a_{ij} x^i (1-x)^j$ , where the coefficients satisfy $a_{ij} \in [0,1]$. Observe that $f(x)$ converges for $x\in [0,1]$. Is $f$ analytic over ...
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### Solving a tough limit

I am trying to verify for which $z \in \mathbb{R}$ the series $\sum _{n=1}^{\infty } \left(1-\cos \left(\frac{1}{n}\right)\right)^z$ converges. The only test that was successful for me is the Kummer ...
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### Can a norm only be defined on vector spaces?

All examples I have come across so far deal with norm defined on a vector space. Can norm only be defined on vector spaces?
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### Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers

A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with non-...
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### Does the sum of reciprocals of primes converge?

Is this series known to converge, and if so, what does it converge to (if known)? Where $p_n$ is prime number $n$, and $p_1 = 2$, $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
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### Is the natural log of n rational?

It's famously unknown whether the natural log of 2 is rational or not. How about the natural log of other numbers. Is it known/unknown whether these are rational? Obviously ln(1) is 0, and ln(2^n) ...
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### Reverse Taylor Series

Almost everyone is familiar with the famous Taylor Series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$ which, if it converges at more than one point, will converge in some interval ...
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### Convergence tests for improper multiple integrals

For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the ...
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Suppose $z$ is a complex number. Prove that there exists an $r \geq 0$ and a complex number $w$ (with $|w| = 1$) such that $z = rw$. Does $z$ uniquely determine $r$ and $w$? Let $z = a+bi$. Then $|... 2answers 256 views ### Complex Numbers and Square Roots Suppose$z = a+bi$and$w = u+iv$. Let$\displaystyle a = \left(\frac{|w|+u}{2} \right)^{1/2}$and$\displaystyle b = \left(\frac{|w|-u}{2} \right)^{1/2}$. Show that$z^2 = w$if$v \geq 0$and$(\bar{...
Fix $b>1, y>0$ and prove that there is a unique real $x$ such that $b^x = y$. So for uniqueness, $x_1 < x_2 \Rightarrow b^{x_1} < b^{x_2}$. Then consider the set $S = \{b^t: t \leq x \}$ ...