It has been a while since I have done any work with infinite series but could someone explain why this is, or offer a proof. (The most similar thing I could think of is the proof of $.9$ repeating equaling $1$)
I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
2 Infinite numbers do exist in the hyperreal number system which properly extends the real number system, but then their reciprocals are infinitesimals rather than zero. Thus the idea of $\frac {1} {0}$ can be interpreted as saying that if $\epsilon$ is infinitesimal then $\frac {1} {\epsilon}$ is infinite.
sequences and series - What is the sum of an infinite resistor ladder ...
If I understand it correctly then, I can't literally take an infinite collection of terms and add them up but by how the sum of the sequence is defined, I can figure out what the the sum approaches as number of terms goes to infinity and can rightfully say that that value (the limit) is the sum of the sequence?
Informally, I would like to find an infinite product of rational numbers that evaluates to a nonzero rational number such that the multiplicity of each prime in the numerator is finite, while on the denominator there are an infinite number of primes with unbounded multiplicity.
Existence of an infinite product that converges to a rational number ...
Recently, I encountered a problem about infinite series. So my question is how to know whether the infinite series $\sum _ {n=2}^ {\infty } \frac {1} {n \log (n)}$ is convergent?