Proof of sum results

I was going through some of my notes when I found both these sums with their results x0+x1+x2+x3+…=11−x,|x|<1 0+1+2x+3×2+4×3+…=1(1−x)2 I tried but I was unable to prove or confirm that these results are actually correct, could anyone please help me confirm whether these work or not? Answer 1−xn+11−x=1+x+x2+⋯+xn, now if n→∞ and |x|<1 we get … Read more

Limits of transfinite numbers

Does it makes sense to talk about: $$ \lim_{i\to \aleph_0} \aleph_i $$ What type of infinity does it approach? Maybe finding a limit of that doesn’t make sense. What about $\aleph_{\aleph_0}$? What type of infinity is that? Answer Yes it makes perfect sense. And $\aleph$ “commutes with lim” so that $$ \begin{align} \lim_{n\to \aleph_0} \aleph_n … Read more

Calculus Improper Integral Convergence; Which is right: Limits or Areas?

Could someone please explain to me the following doubt I have on improper integral: ∫∞−∞1x dx I still think that since integrals signify areas that this evaluates to 0 instead of DNE (Does not Exist). I understand that limit notation solution for this improper integral returns DNE, but the areas from −∞ to ∞ cancel. Answer … Read more

What happens when $r \to \infty$? Will it be a line? (partial circle)

Let $a$ be a arc of particle circles, which is constant. What happens when $r \to \infty$? Will it be a line? Radius of partial circle : $r$, Arc of partial circle : $a$ and constant, For $r=r_0$ For $r\simeq 2.r_0$ For $r>>>r_0$, like a line, For ,$\quad\lim\limits_{r\to \infty}\quad$ ,$\quad a$ ,can be partial line? … Read more

Why does this limit equal 0?

I’m trying to solve the limit: limx→∞x3+2xx2+3x So far I’ve divided every term by 3^x and this is correct according to the solution, which gets me: \lim_{x \to \infty}\frac{\frac{x^3}{3^x}+( \frac{2}{3})^x}{\frac{x^2}{3^x}+1} I would’ve said that the X^3/3^x goes to 0 and the (2/3)^x goes to infinity and then x^2/3^x goes to 0 and the one stays … Read more

How to prove that infinity (∀n∈N:x>n\forall n \in \mathbb N: x>n) is not a real number?

By infinity, I mean a number x, such that x>n for any natural number n. Here, x+1 would just be x+1, so we couldn’t use something such as x+1=x>x to show that it doesn’t exist. Its multiplicative inverse, ϵ, would also be a real number, smaller than any other positive real number. My first thought … Read more

A Question on Cardinality ℵ0\aleph_{0}

I’m trying to understand the concept of Cardinality. My question is, Let the interval [1,2n] is given. In this interval we have 2n natural numbers. Or n→∞, we have countable infinite natural numbers and Cardinality equal to ℵ0. Then, in this interval we have n even natural numbers. Or n→∞, we have countable infinite even … Read more

Does infinity cause incompleteness in formal systems? Is a finite formal system complete?

Like most, I’m having a hard time understanding the consequences of Gödel’s Incompleteness Theorems. In particular, I’d like to understand their connection to the concept of infinite mathematical structures. In doing so, I hope to formulate a better opinion on the merits of constructivism and finitism in regards to Gödel’s theorems. Without being philosophical, I … Read more