What is the definition of the absolute convergence of an infinite product (1+a_1)(1+a_2)(1+a_3)\cdots(1+a_1)(1+a_2)(1+a_3)\cdots?

For an infinite product (1+a_1)(1+a_2)(1+a_3)\cdots, whats the definition of convergence and absolute convergence? Why the absolute convergence corresponding to the absolute convergence of sum of an infinite series a_1+a_2+a_3+\cdots? If the infinite product absolutely convergent, does it mean it is convergent? If the sum of a_1+a_2+a_3+\cdots convergent, does it mean the corresponding product convergent? I … Read more

Constructing a smooth function whose roots consist only of each of the primes.

My first attempt: f(x)=∞∏i=1(1−xpi) If we take a look at the Riemann zeta function: ζ(s)=∞∑n=11ns=∞∏i=0(11−p−si)=∞∏i=0(1−1psi)−1 f(1)=1ζ(1)=0 By f‘s construction, it should only contain 0 factors at prime xs, which 1 is not. Therefore, the only reason f should be 0 at 1 is that the product converges to 0 as i→∞. My second attempt: g(x)=∞∏i=1(1−x2p2i) … Read more

Prove that ∞∏n=k0(1−an){\prod_\limits{n = k_0}^{\infty} (1 – a_n)} \; converges to positive value

Request help proving the following: Given: (1) For all n>0,an>0 (2) ∞∑n=1an is convergent. (3) There exists k0∈Z+ such that for all k≥k0,ak<1. To prove: ∞∏n=k0(1−an) converges to a λ that is strictly greater than 0. Motive: On pages 10-11 of this pdf, the assertion helps prove that if ∞∑n=1an is convergent, then the infinite … Read more

What’s the sum of the inverses of the Primorial numbers?

What’s the sum of the inverses of the primorial numbers? Let the $n^{th}$ primorial number be the product of the first $n$ primes $\displaystyle n\#= \prod_{p\leq p_n}p$ So $N\#=2,2\cdot3,2\cdot3\cdot5,\ldots=2,6,30,210,\ldots$ Evaluate $\displaystyle\sum_{n\in\Bbb N}\frac1{n\#}$ Here’s what very limited part of this I can do: Obviously it’s in the fairly narrow interval $(\frac23,e-2)$ by comparing the first two … Read more

Limit point pp of A⊂2RA \subset 2^\mathbb{R} such that no sequence in AA converges to pp. Can AA be countable?

The problem here is to find a subset A⊂2R and a limit point p of A such that no sequence in A converges to p. Here, 2R is the product ∏λ∈R{0,1} equipped with the product topology. I think I have an example for when A is not countable. Suppose A is the set of all … Read more

Landau’s proof that ζ(s)=∏p11−p−s\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}

In the Handbuch, Landau proves that for all s>1 the following equality holds ζ(s)=∏p11−p−s. I’m having trouble with the following part of his proof: Landau says that for all s>1 we have ∏p≤x11−p−s=∏p≤x(1+1ps+1p2s+…)=∞∑n=1′1nswhere in the last sum n runs through all numbers whose prime factors are all ≤x. Can someone please shed light on the … Read more

Infinite Product – Seems to telescope

Evaluate (1+23+1)(1+232+1)(1+233+1)⋯ It looks like this product telescopes: the denominators cancel out (except the last one) and the numerators all become 3. What would my answer be? Answer we have the following identity (which affirms that the product telescopes): (1+23n+1)=3⋅3n−1+13n+1=1+3−(n−1)1+3−n (as denoted in the comment by Thomas Andrews)and as a result: n∏k=1(1+23k+1)=2⋅3n3n+1=21+3−n AttributionSource : Link … Read more

Limit of a product I

While reviewing old problems in American Mathematical Monthly the following problem was encountered. What are some methods to solving the problem ? Proposed by L. S. Johnston, 1929. Consider the infinite sequence $\{ a_{n} \}$ of real positive numbers with the recurrent relation \begin{align} a_{k+1}^{2} = \frac{2 \, a_{k}}{a_{k} + 1} \end{align} for $k \geq … Read more

Equivalence of convergence of a series and convergence of an infinite product

Let (an)n be a sequence of non-negative real numbers. Prove that ∞∑n=1an converges if and only if the infinite product ∞∏n=1(1+an) converges. I don’t necessarily need a full solution, just a clue on how to proceed would be appreciated. Could any convergence test like the ratio or the root test help? Answer Hint: Since the … Read more

Can we approximate $\prod_{n=1}^\infty (1-\frac{(2n+1)x^2}{n^2\pi^2})$?

It seems like from the graph $\prod_{n=1}^\infty (1-\frac{(2n+1)x^2}{n^2\pi^2})$ is somehow alike to the graph $e^{-x^2}$, the main problem is that the limitations of the software makes it hard to graph for large numbers. Is it possible to do so or it diverges? Answer The partial product $$P_m=\prod _{n=1}^{m } \left(1-\frac{(2 n+1) }{n^2\pi ^2 }x^2\right)$$ effectively … Read more