Talk:Arithmetic progression

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[edit] Merging with arithmetic series

According to Mathworld, which has a link in the article, an arithmetic series is the sum of an arithmetic progression or sequence. Charles Matthews has obscured this distinction by redirecting arithmetic series to arithmetic progression. I'm not sure whether the distinction made by Mathworld is commonly recognised by mathematicians, so I'm not going to revert the change. I'll wait for comments. -- Heron 15:12, 7 Mar 2004 (UTC)

The redirect only implies that information on arithmetic series is contained in the arithmetic progression article, not that the two terms are synonymous. The article is quite clear on the latter point (and mathworld is right, of course). -- Arvindn 15:39, 7 Mar 2004 (UTC)

Yes, the article has both definitions; I don't see anything obscure about it.

Charles Matthews 15:59, 7 Mar 2004 (UTC)

How about merging series&progression articles as it's done with the geometric series&progression?

I strongly think arithmetic series should be merged back into this article. Fredrik | talk 14:20, 19 August 2005 (UTC)
Fredrik, thanks! Oleg Alexandrov 19:35, 20 August 2005 (UTC)
No problem :) - Fredrik | talk 19:38, 20 August 2005 (UTC)

[edit] Product

I toyed with the idea of taking the product of an arithmetic progression, and came up with the following expression (initial term a, common distance s, and n terms):

s^n \times \frac{\Gamma \left(a/s + n\right) }{\Gamma \left( a / s \right) }

Anyone seen this before, and is it useful? - Fredrik | talk 16:38, 19 August 2005 (UTC)

Interesting. This seems to be generalizing the formula 1·2··· n=Γ(n+1). I hever saw it before. I quite don't know what one would use it for though.
This article is rather stubby, any additions to it, like your formula, would be welcome.
By the way, what do you think of merging this article with Arithmetic series? Both are stubby and don't talk about much different things? Oleg Alexandrov 19:09, 19 August 2005 (UTC)
Yeah, it's derived from that formula.
It could be useful in numeric computation, to obtain the product (or its logarithm) of an immensely long progression in O(1) time, though I'm not sure in what kind of context you'd need to do that.
There is also an obvious problem, that it is invalid when a/s is a negative integer (though for computations that could be handled easily as a special case).
As stated a couple of paragraphs up, yes, I think merging would be a good idea. Fredrik | talk 19:35, 19 August 2005 (UTC)
So we arrived independently to the same conclusion. I will merge the articles soon if I don't forget. If you get to it before me, that will be fine too. Oleg Alexandrov 03:53, 20 August 2005 (UTC)
Actually, seems like I'd just rediscovered the Pochhammer symbol, heh. Fredrik | talk 13:25, 23 October 2005 (UTC)

[edit] sum of sine and cosine in arithmetic progression

I think this would be useful to add under 1 Sum (arithmetic series)

[edit] Sum of Sines

The arguments of a sum of sines can be in arithmetic progression, as follows

S=\sin \varphi + \sin {(\varphi + \alpha)} + \cdots + \sin {(\varphi + n\alpha)}.

It has also, like a normal arithmetic sequence, a concise formula, written as(mitchell lanser is gay)

S=\frac{\sin{(\frac{(n+1) \alpha}{2})} \cdot \sin{(\varphi + \frac{n \alpha}{2})}}{\sin{\frac{\alpha}{2}}}.

[edit] Sum of Cosines

Analogous to the sum of sines with their arguments in arithmetic sequence, there is also one with cosines:

S=\cos \varphi + \cos {(\varphi + \alpha)} + \cdots + \cos {(\varphi + n\alpha)}.

There is also, the general expression, which is somewhat similar to the one of the sines:

S=\frac{\sin{(\frac{(n+1) \alpha}{2})} \cdot \sin{(\varphi + \frac{n \alpha}{2})}}{\sin{\frac{\alpha}{2}}}
S=\frac{\sin{\frac{(n+1) \alpha}{2}} \cdot \cos{(\varphi + \frac{n \alpha}{2})}}{\sin{\frac{\alpha}{2}}}.
Anyone give me some input on anything else to write, I hope this section can be improved before posted on the article itself.

Cako 21:20, 2 November 2006 (UTC)