Wikipedia:Reference desk/Archives/Mathematics/2008 June 19

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June 19

Infinite Series Involving Prime Numbers

I came across the following series while studying some Fourier Analysis. Given the infinite series

$S=1+{\frac {1}{5}}-{\frac {1}{7}}-{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}-{\frac {1}{19}}-{\frac {1}{23}}+\cdots$

After writing it in the following closed form

$S={\frac {6}{5}}+\sum _{k=2}^{\infty }\left((-1)^{k-1}\right)\left({\frac {1}{p(2k)}}+{\frac {1}{p(2k+1)}}\right)$

where $p(n)$ returns the n-th prime number. Now this series seems to be converging and in fact it seems to be converging to a numerical value of $S=1.046932...$. My question is, how can we tell for sure if this series is converging or not and if it is converging, what is it converging to? Is there anyway to find out what the exact value is? Is this a well-known fact, a well-known constant?A Real Kaiser (talk) 01:49, 19 June 2008 (UTC)

The series certainly converges, by the alternating series test. Have you tried Plouffe's inverter for the limit (I don't have a precise enough estimate)? Algebraist 01:13, 19 June 2008 (UTC)

Actually no, I had not tried looking up in the inverter. I want to but the problem is that this series converges so slowly, after adding up the first 100,000 terms, I still only had 1.04693... and I am not even sure about the last digit 2 yet. So, first I have to write up a more efficient program to calculate more digits (faster) so that I can obtain more decimal places. Meanwhile, does anyone have any idea what this number is?A Real Kaiser (talk) 01:49, 19 June 2008 (UTC)

I'm fairly certain it starts 1.046932971... (with the last 1 possibly a 0). Plouffe's returns no hits. -- Meni Rosenfeld (talk) 09:56, 19 June 2008 (UTC)

See Proof that the sum of the reciprocals of the primes diverges -- Q Chris (talk) 10:32, 19 June 2008 (UTC)

How is this relevant? -- Meni Rosenfeld (talk) 10:47, 19 June 2008 (UTC)

I think I've found it on OEIS: [1] - note that their formulation differs from yours, because you've omitted 2 and 3, and included 1. Oops, didn't notice the oddity in your thing. It isn't the same after all. Well, maybe that's a start, at least.--Fangz (talk) 11:35, 19 June 2008 (UTC)

Maths (Indian mathematicians)

How do I get the clips of Sridhara on wikipedia?Minintelligent (talk) 10:52, 19 June 2008 (UTC)

What do you mean by "clips"? You can add pieces of information to the Sridhara article if you have anything suitable. If you are having trouble with editing then you should try reading Wikipedia:How to edit a page, which gives lots of assistance. --_{tiny plastic} Grey Knight ⊖ 11:55, 19 June 2008 (UTC)

Meaning of Exponent In Trigonometric Expressions?

What does $\cos ^{2}(x)$ mean? And $\sin ^{2}(x)?$ And $\tan ^{2}(x)$? Etc.
The Piano Man (talk) 14:14, 19 June 2008 (UTC)

• $\sin ^{2}(x)$ means $(\sin(x))^{2}$ or equivently $\sin(x)*\sin(x)$. It is used to distinguish easily from $\sin(x^{2}))$. Similarly the same rule applies for the other trig functions. Rambo's Revenge (talk) 14:32, 19 June 2008 (UTC)

It's important to note that for most other functions, the same notation means to apply the function twice, $f^{2}(x)=f\circ f(x)=f(f(x)$. The way it's used for trig function is an abuse of notation, but it's a very convenient one. --Tango (talk) 14:39, 19 June 2008 (UTC)

Be careful though, as $\sin ^{-1}(x)\neq {\dfrac {1}{\sin(x)}}$, but is instead equal to $\arcsin(x)$. -mattbuck (Talk) 15:20, 19 June 2008 (UTC)

And if you see $\sin ^{-2}(x)$, it's best just to run away and hide! --Tango (talk) 15:27, 19 June 2008 (UTC)

Are you sure about that? I’ve only seen this notation used as composition when $f$ is being considered as an operator. It then only naturally makes sense because operator spaces have their multiplication operation as composition, while function spaces have their multiplication operation as pointwise multiplication. Thus if $f$ is being considered an operator, it is unambiguous $f^{2}(x)=(f\cdot f)(x)=f(f(x))$, while if $f$ is being considered in a typical function space it is still unambiguous, $f^{2}(x)=(f\cdot f)(x)=(f(x))^{2}$ GromXXVII (talk) 00:02, 20 June 2008 (UTC)

The notation is sometimes unambiguous for trivial reasons: if the image of f is not contained in its domain, then the first interpretation is impossible, while if the range of f has no multiplication operation, then the second is impossible. Algebraist 00:09, 20 June 2008 (UTC)

Yeah, pretty sure. That's the way I've generally seen it used. The most obvious example is f^-1 - that almost universally means the inverse of f, never it's reciprocal, and other exponents generally work the same way. I'm not sure what you mean by a "function space" - I would usually that term to refer to a vector space of functions, which doesn't have a multiplication (other than scalar multiplication, of course). The closest you get to a multiplication would be an inner product, but that gives you a scalar, not a function. --Tango (talk) 12:14, 20 June 2008 (UTC)

I think he was referring to when you have an algebra of functions. --137.205.233.134 (talk) 13:06, 20 June 2008 (UTC)

savant mathématicien

Bonjour, Pourquoi M.Jan MIKUSINSKI né le 3 avril 1913 à Stanislawow et décédé le 27 juillet 1987 à KATOWICE ne figure pas dans la liste des grands mathématiciens connu pour son travail de pionnier dans l'analyse mathématique.M.Mikusinski a développé un calcul opérationnel 44a40 calculus of Mikusinski,qui sont pertinentes pour résoudre les équations différentielles.Son calcul opérationnel est basé sur un calcul de la convolution des fonctions à l'égard de la transformés de Fourier.De la convolution produit qu'il va à définir ce que dans d'autres contextes est appelé le domaine de fractions ou un quotient domaine.Ces couples de fonctions M.Mikusinski appels opérateurs-Mikusinski opérateur,44a40 il est également bien connu pour Cube et Antosik-Mikusinski Théorem,Mikusinski convolution algèbre etc. Récupéré "de http://en.wikipedia.org/wiki/Jan_Mikusinski" Plusieurs grands savants polonais figurent aux États-Unis en Suède et font des conférences partout dans le monde. La Pologne peut être très fière d'avoir des savant d'une telle renommée dans le monde entier. —Preceding unsigned comment added by 90.54.102.79 (talk) 14:41, 19 June 2008 (UTC)

Sir, this is not a question, but a statement. You might have also read that he died twenty years ago... Monsieur, ce n'est pas une question, mais une déclaration. Vous avez peut-être également lire qu'il est mort il ya vingt ans. 86.152.76.21 (talk) 17:02, 19 June 2008 (UTC)

Ne pleure pas, Alfred - j'ai besoin tout mon courage pour mourir á vingt ans! Damien Karras (talk) 21:05, 19 June 2008 (UTC)

I don't understand this language. Can anyone make a translation?--218.102.124.108 (talk) 17:03, 20 June 2008 (UTC)

It's french. He's complaining that a certain man did various work in mathematical analysis, but does not figure in some list of great mathematicians.--Fangz (talk) 23:13, 20 June 2008 (UTC)

Translation: Hello, Why does Mr. Jan MIKUSINSKI born April 3 1913 in Stanislawow, deceased July 27 1987 in KATOWICE and known for his pioneering work in mathematical analysis not figure in the list of great mathematicians[?] Mr. Mikusinski developed an operational calculus [which helps] resolve differential equations. His calculation...is based on the convolution of Fourier functions. The convolution produces what is defined in other contexts as the domain of fractions or a quotient domain. Mr. Mikusinski calls this set of operators mikusinski operater, it is known equally well as Cube and antosik-mikusinski theorem, Mikusinski convolution algebra, etc. Recovered from http://en.wikipedia.org/wiki/Jan_Mikusinsk . Several great Polish scholars figure in the United States [and] in Sweden and give lectures all over the world. Poland can be very proud to have such world famous scholars. Cuddlyable3 (talk) 14:30, 21 June 2008 (UTC)