Translation please,
 

[URL]https://www.wolframalpha.com/input/?i=product+n%5Ei%2Bn,+i%3D1+to+k[/URL]


* What is the semicolon?

* How does this expand to:

2n(n[SUP]2[/SUP]+n)(n[SUP]3[/SUP]+n)...(n[SUP]k[/SUP]+n)

Thank you in advance.:smile:

Doesn't it say right there?
the [I]q[/I]-Pochhammer symbol (a; q)[SUB]n[/SUB]

[QUOTE=Batalov;463441]Doesn't it say right there?
the [I]q[/I]-Pochhammer symbol (a; q)[SUB]n[/SUB][/QUOTE]
Good eyes.
Thank you.

There is sigma as the symbol for sum of series, there is the equivalent symbol for product of series ( what is that symbol called? ).

Is there an equivalent symbol/operator for the power of series?

Replacing the word "sum" or "product" with "power" in Wolfram alpha does not return expected results.

Thank you in advance.

[QUOTE=a1call;463463]There is sigma as the symbol for sum of series, there is the equivalent symbol for product of series ( what is that symbol called? ).
[/QUOTE]

[URL="https://en.wikipedia.org/wiki/Pi_%28letter%29"]Capital pi.[/URL]

[QUOTE=paulunderwood;463464][URL="https://en.wikipedia.org/wiki/Pi_%28letter%29"]Capital pi.[/URL][/QUOTE]



:blush:Doh:blush:

Correct me if I am wrong, but it should be possible to represent the power-of-series using iteration notation.
[url]https://en.m.wikipedia.org/wiki/Iterated_function[/url]

Would appreciate confirmation or negation.

Thank you in advance.

The concept seems absent on the net:

[url]http://www.google.com/search?q=%22power+of+series%22[/url]

[QUOTE=a1call;463476]The concept seems absent on the net:

[url]http://www.google.com/search?q=%22power+of+series%22[/url][/QUOTE]

are we talking geometric series, arithmetic series, harmonic series?, or power series? are they finite, or infinite ? I once saw a math.stackexchange question about powers using arithmetic progressions in order for example ( not quite the same but interesting non the less).

[QUOTE=science_man_88;463480]are we talking geometric series, arithmetic series, harmonic series?, or power series? are they finite, or infinite ? I once saw a math.stackexchange question about powers using arithmetic progressions in order for example ( not quite the same but interesting non the less).[/QUOTE]

[url]https://www.wolframalpha.com/input/?i=sum+n%5Ei%2Bn,+i%3D1+to+k[/url]

[url]https://www.wolframalpha.com/input/?i=product+n%5Ei%2Bn,+i%3D1+to+k[/url]

[url]https://www.wolframalpha.com/input/?i=power+n%5Ei%2Bn,+i%3D1+to+k[/url]

?= (n^1+n)^(n^2 +n)^.... [k times]

[QUOTE=a1call;463481][url]https://www.wolframalpha.com/input/?i=sum+n%5Ei%2Bn,+i%3D1+to+k[/url]

[url]https://www.wolframalpha.com/input/?i=product+n%5Ei%2Bn,+i%3D1+to+k[/url]

[url]https://www.wolframalpha.com/input/?i=power+n%5Ei%2Bn,+i%3D1+to+k[/url]

?= (n^1+n)^(n^2 +n)^.... [k times][/QUOTE]

iterated functions are [TEX]f^n(x)[/TEX] so like the functions of LL or [TEX]M_n [/TEX]

for LL it's :

[TEX]f(x)=x^2-2 \ \pmod {M_p}[/TEX] but starting with x=4

for generating the next mersenne number you can use:

[TEX]f(x)=2x+1[/TEX] starting at x=0 ( or x=1 depending on what you think of 0 being a mersenne number, under one of the definitions) . I guess, I don't get where you are going.

Not really going anywhere. Just sitting on my balcony, exercising my mind.:smile:

It's odd no one seems to have contemplated the concept of power-of-series before.

[QUOTE=a1call;463484]Not really going anywhere. Just sitting on my balcony, exercising my mind.:smile:

It's odd no one seems to have contemplated the concept of power-of-series before.[/QUOTE]

the result would depend on if they were finite or infinite etc.

That is the case with all series operations including sum and product. That's no reason for such a concept not to exist.

[QUOTE=a1call;463486]That is the case with all series operations including sum and product. That's no reason for such a concept not to exist.[/QUOTE]

sorry I'm probably confusing you I was thinking arithmetic progressions etc. but a series is a sum. so what is the product of two sums ? and I think you might look up the multinomial theorem.

A series is a number of progressive terms (finite or else).
The sum-of-series is the sum of the terms.
The product-of-series is the product of the terms.
The power-of-series would be the power of first term to power of the second term....
There is a similar concept named tower, but not quite the same thing.

[QUOTE=a1call;463488]A series is a number of progressive terms (finite or else).
The sum-of-series is the sum of the terms.
The product-of-series is the product of the terms.
The power-of-series would be the power of first term to power of the second term....
There is a similar concept named tower, but not quite the same thing.[/QUOTE]

technically a [URL="https://en.wikipedia.org/wiki/Series_(mathematics)"]series[/URL] is a sum of terms.

[QUOTE=science_man_88;463489]technically a [URL="https://en.wikipedia.org/wiki/Series_(mathematics)"]series[/URL] is a sum of terms.[/QUOTE]
Thank you for the clarification. I find that definition illogical.
I suspect it is a relatively recent definition for the word, hence the inclusion of the term "roughly" in that article.

[QUOTE=a1call;463490]Thank you for the clarification. I find that definition illogical.
I suspect it is a relatively recent definition for the word, hence the inclusion of the term "roughly" in that article.[/QUOTE]

I think roughly is to simplify the technical meaning. Taylor series for example, go back to a mathematician from the late 1600's early 1700's.

Here they distinguish the term series and sequence, which to me are synonimous.
Note the conflicting title:

[url]https://www.varsitytutors.com/hotmath/hotmath_help/topics/sum-of-the-first-n-terms-of-a-series[/url]


[url]http://www.google.com/search?q=sequence+synonimous[/url]

[QUOTE]se·quence
ˈsēkwəns/
noun
1.
a particular order in which related events, movements, or things follow each other.
synonyms: succession, order, course, [B]series[/B], chain, train, string, progression, chronology, timeline; More[/QUOTE]

[QUOTE=a1call;463492][url]http://www.google.com/search?q=sequence+synonimous[/url][/QUOTE]

[QUOTE][B]MATHEMATICS[/B]
an infinite ordered series of numerical quantities.[/QUOTE]

so series is sometimes used to describe it even in math . meaning is always context dependent.

[url]http://www.purplemath.com/modules/series.htm[/url] may help you.

Sequence is 1,2,3,...; series is 1+2+3+....

[QUOTE=CRGreathouse;463539]Sequence is 1,2,3,...; series is 1+2+3+....[/QUOTE]

* I wonder, how true that is languages other than English. I am pretty sure the series equivalent in Persian which is pronounced as "sery" and is probably introduced from French, does not have the summation concept included, even in Mathematical context.

* I wonder, How recent is the summation notion associated with the word "Series" in English. It is safe to presume that the word "Series: predates the concept of summation of Series/sequences. I would put my money on only decades old at most.

[QUOTE=a1call;463548]* I wonder, how true that is languages other than English. I am pretty sure the series equivalent in Persian which is pronounced as "sery" and is probably introduced from French, does not have the summation concept included, even in Mathematical context.

* I wonder, How recent is the summation notion associated with the word "Series" in English. It is safe to presume that the word "Series: predates the concept of summation of Series/sequences. I would put my money on only decades old at most.[/QUOTE]

You know what they say about a fool, an their money ... From the wikipedia on series ( mathematics), "This paradox was resolved using the concept of a limit during the 19th century." aka the 1800's had infinite series paradoxes solved. The word series, meaning row, or chain, and spelled the modern way was from the 17th century, look at the etymology on Google. The development of calculus, in it's modern form, would not be possible without them.

[QUOTE="http://jeff560.tripod.com/s.html"]SERIES. According to Smith (vol. 2, page 481), "The early writers often used proportio to designate a series, and this usage is found as late as the 18th century."

John Collins (1624-1683) wrote to James Gregory on Feb. 2, 1668/1669, "...the Lord Brouncker asserts he can turne the square roote into an infinite Series" (DSB, article: "Newton").

James Gregory wrote to John Collins on Feb. 16, 1671 [apparently O. S.]: "I do not question that all equations may be formed by tables, but I doubt exceedingly if all equations can be solved by the help only of the tables of logarithms and sines without serieses."

According to Smith (vol. 2, page 497), "The change to the name ’series' seems to have been due to writers of the 17th century. ... Even as late as the 1693 edition of his algebra, however, Wallis used the expression 'infinite progression' for infinite series."

In the English translation of Wallis' algebra (translated by him and published in 1685), Wallis wrote:

Now (to return where we left off:) Those Approximations (in the Arithmetick of Infinites) above mentioned, (for the Circle or Ellipse, and the Hyperbola;) have given occasion to others (as is before intimated,) to make further inquiry into that subject; and seek out other the like Approximations, (or continual approaches) in other cases. Which are now wont to be called by the name of Infinite Series, or Converging Series, or other names of like import.[/QUOTE]

[QUOTE=a1call;463486]That is the case with all series operations including sum and product. That's no reason for such a concept not to exist.[/QUOTE]
The distinction is that taking powers is not associative.

[QUOTE=Nick;463583]The distinction is that taking powers is not associative.[/QUOTE]

It is indeed a distinction, but not a disqualifier. Same way it does not disqualify exponentiation as a valid function.

[url]https://math.stackexchange.com/questions/2351850/power-towers-and-notation-for-iterated-exponentiation[/url] might be of interest to a1call

[QUOTE]Sorry science_man_88 is a moderator/admin and you are not allowed to ignore him or her.[/QUOTE]

What a shame.

[QUOTE=a1call;463594]What a shame.[/QUOTE]

I would have said sham I might as well be brown as it could signal full of :poop: the blog area was mostly made as a trash bin at last check.

The real source of sm88's quotes is Jeff Miller's [url=http://jeff560.tripod.com/mathword.html]Earliest Known Uses of Some of the Words of Mathematics[/url], section [url=http://jeff560.tripod.com/s.html]S[/url], which indeed dates it to the 17th century. By 1700 the terminology was nearly universal, so it caught on quickly -- just a generation or so.

[QUOTE=a1call;463484]It's odd no one seems to have contemplated the concept of power-of-series before.[/QUOTE]

[QUOTE=a1call;463486]That is the case with all series operations including sum and product. That's no reason for such a concept not to exist.[/QUOTE]

You might find Knoebel's [url=https://www.jstor.org/stable/2320546]Exponentials Reinterated[/url] (which won two prizes for expository writing) interesting. I don't think it covers your exact case but you can probably use its methods to prove results about it.

[QUOTE=CRGreathouse;463612]The real source of sm88's quotes is Jeff Miller's [url=http://jeff560.tripod.com/mathword.html]Earliest Known Uses of Some of the Words of Mathematics[/url], section [url=http://jeff560.tripod.com/s.html]S[/url], which indeed dates it to the 17th century. By 1700 the terminology was nearly universal, so it caught on quickly -- just a generation or so.[/QUOTE]
Firstly, that is an amazing source. Secondly you are an amazing researcher.
Thank you.

However, I could not decipher the origin of association of summation in those references with the word series or its variations.
Would be willing to eat my hat (if I was wearing one:smile:) if otherwise shown.

[QUOTE=CRGreathouse;463612]The real source of sm88's quotes is Jeff Miller's [url=http://jeff560.tripod.com/mathword.html]Earliest Known Uses of Some of the Words of Mathematics[/url], section [url=http://jeff560.tripod.com/s.html]S[/url], which indeed dates it to the 17th century. By 1700 the terminology was nearly universal, so it caught on quickly -- just a generation or so.[/QUOTE]

on that same site there's :


[URL="http://jeff560.tripod.com/mathsym.html"]Earliest Uses of Various Mathematical Symbols[/URL]

and

[URL="http://jeff560.tripod.com/ambiguities.html"]Ambiguously Defined Mathematical Terms
at the High School Level[/URL]

as well as :

[URL="http://jeff560.tripod.com/stamps.html"]Images of Mathematicians on Postage Stamps[/URL]

seems an interesting set of links.

FWIW:

Definition of geometric series

[QUOTE]First Known Use: circa 1909
[/QUOTE]

[url]https://www.merriam-webster.com/dictionary/geometric%20series[/url]


series
[QUOTE]First Known Use: 1611[/QUOTE]

[url]https://www.merriam-webster.com/dictionary/series[/url]

[QUOTE=a1call;463648]FWIW:

Definition of geometric series



[url]https://www.merriam-webster.com/dictionary/geometric%20series[/url]


series


[url]https://www.merriam-webster.com/dictionary/series[/url][/QUOTE]

there are many types of series so what's the point ? there are infinite series, finite series, convergent series, divergent series, formal power series, power series, geometric series, arithmetic series, harmonic series, basically if there's a type of sequence, you can turn it into a type of series. This list also forgets special kinds like Taylor series, Maclaurin series, etc. Edit:

[QUOTE="http://jeff560.tripod.com/g.html"]GEOMETRIC SERIES is found in 1723 in A System of the Mathematics James Hodgson [Google print search, James A. Landau].[/QUOTE] though it may be in a different definition who knows.

[B][SIZE=3]Webster's Dictionary 1828 - Online Edition[/SIZE][/B]



[QUOTE][B]Series[/B]



...

[B]4.[/B] In [I]arithmetic[/I] and [I]algebra[/I], a number of terms in succession, increasing or diminishing in a certain ratio; as arithmetical [I]series[/I] and geometrical [I]series[/I]. [See [I]Progression[/I].]
[/QUOTE]

In 1913:

[QUOTE][B]3.[/B] [I](Math.)[/I] An indefinite number of terms succeeding one another, each of which is derived from one or more of the preceding by a fixed law, called the [I]law[/I] of the series; as, an arithmetical [I]series[/I]; a geometrical [I]series[/I].
[/QUOTE]



[url]http://www.websters1913.com/words/Series[/url]

[QUOTE=a1call;463650][B][SIZE=3]Webster's Dictionary 1828 - Online Edition[/SIZE][/B]





In 1913:





[url]http://www.websters1913.com/words/Series[/url][/QUOTE]

okay, well I believe these sequences of numbers, are usually the easiest nowadays to find the value of the sum, if they fit within certain rules. [TEX]a\over1-r[/TEX] a being teh starting term and r being the ratio for a geometric sequence being summed as long as this is not greater in absolute value than 1.

Websters Definitions in 1943 are attached.

[QUOTE=a1call;463653]Websters Definitions in 1943 are attached.[/QUOTE]

I agree see Succession. in theory succession could be adding.