Talk:Sophomore's dream

Cool!

Title added. —Nils von Barth (nbarth) (talk) 06:32, 12 February 2011 (UTC)

Cool!

(But this article is not linked-to enough. I'll see if I can do something about that.) Michael Hardy 23:36, 6 February 2007 (UTC)

Why "sophomore's dream"?

Title added. —Nils von Barth (nbarth) (talk) 06:32, 12 February 2011 (UTC)

But why "sophomore's dream"? 24.137.126.62 09:08, 28 February 2007 (UTC)

Agreed... a reason for the name would be good to add. Privong 04:20, 16 March 2007 (UTC)

I'm wondering the same thing. There's probably an interesting story behind it. Roger 18:11, 14 May 2007 (UTC)

I came to the talkpage to see if this was being discussed actually. The etymology would add to the article. Mehmet Karatay 13:00, 20 June 2007 (UTC)

I agree that some etymology would flesh this out a bit. —Preceding unsigned comment added by 143.44.71.171 (talk) 01:28, 1 October 2007 (UTC)

I added something, but I couldn't find an adequate reference so at the moment it's unsourced. skeptical scientist (talk) 05:18, 12 May 2008 (UTC)

I don’t know the origin of the name; I’ve listed the earliest reference which is given (2004, as noted on Wolfram, for instance), and this seems to be the origin of the name – any more formal references would be appreciated!

—Nils von Barth (nbarth) (talk) 06:32, 12 February 2011 (UTC)

Expression

Wouldn't the second be simpler as follows?

$\sum _{n=1}^{\infty }-(-1)^{n}n^{-n}=\sum _{n=1}^{\infty }-(-1)^{-n}n^{-n}=-\sum _{n=1}^{\infty }(-n)^{-n}$

I feel like the rightmost expression is much better. --Cheeser1 16:17, 9 July 2007 (UTC)

True, but I think the current expression is in the standard form for an alternating series. Roger 16:25, 9 July 2007 (UTC)

It's not though - it starts at a different index. --Cheeser1 01:39, 1 October 2007 (UTC)

Images

I don't think the graphs of the functions are very accurate - it appears that as x approaches zero, y tends to a value either slightly above or slightly below 1, when the the actual limit should be exactly 1. Maybe someone with good quality graphing software could confirm my thoughts and upload better images? Pscholl (talk) 12:57, 11 May 2008 (UTC)

Agree with Pscholl. It is sufficiently misleading that it would be better to remove these images. AdamWGibson (talk) 05:22, 4 June 2008 (UTC)

I made a more accurate graph showing both functions. If anyone has any suggestions it would not be difficult to fix... Blues shuffle (talk) —Preceding undated comment was added at 01:06, 12 February 2009 (UTC).

Note

The note added to the mention of Freshman's Dream seems hardly relevant. The term Freshman's Dream, in my experience, refers to that equation with real numbers $x$ and $y$, and the arithmetic operations taking on their standard denotations for real numbers, in which case it is, plain and simple, not generally true. I recommend removing the note, but if anyone feels it's necessary to discuss the exceptions that exist in modern abstract algebra I'd like to hear the rationale. 107.5.98.72 (talk) 21:59, 18 June 2011 (UTC)

I think the note is useful, if only to prevent people from changing the text of the article proper to handle that technicality. If the note weren't there, I would have been tempted to change "the incorrect equation" to something like "the (in general) incorrect equation" or "the (over the reals) incorrect equation", both of which are clunkier and less informative. Perhaps a better rewording is possible, though. 24.220.188.43 (talk) 04:19, 3 September 2011 (UTC)

Computing ∫₀¹ xⁿ (logx)ⁿdx .

The way it is currently written is neat and clear, but I think a quicker way (by linking to Euler integral or to Gamma function) wolud be preferable. Precisely, by the change variable $\scriptstyle x=\exp \,{-{\frac {u}{n+1}}}$, with $\scriptstyle 0<u<\infty$, the integral

$\int _{0}^{1}x^{n}(\log \,x)^{n}dx$

writes

$(-1)^{n}(n+1)^{-(n+1)}\int _{0}^{\infty }u^{n}e^{-u}du$

and here we may just put a link to the Euler integral's identity to conclude

$\int _{0}^{\infty }u^{n}e^{-u}du=n!$

Note that the current computation of $\scriptstyle \int _{0}^{1}x^{n}(\log \,x)^{n}dx$ by iterated integration by parts, is exactly what one does for $\int _{0}^{\infty }u^{n}e^{-u}du$, so there is no loss of content if we directly refer to the more popular identity for the Euler integral. --pma 07:48, 9 July 2011 (UTC)

done --pma 17:09, 23 November 2011 (UTC)

As described on Euler integral and Gamma function, shouldn't $\scriptstyle \int _{0}^{\infty }u^{n}e^{-u}du$ come out as $(n+1)!$? --jftsang 19:26, 20 April 2012 (UTC)

No. Look at the linked formulas more carefully. --pma 12:37, 23 April 2012 (UTC)

Thanks so much pma (PMajer) – that makes what’s going on much clearer. (Thanks for the kind words on the earlier description, which I wrote years ago.) I’ve rewritten it a bit (in this edit), making the Gamma function more explicit, and a note explaining the substitution, otherwise it appears a bit magic. I also wrote out an equation to make the steps explicit, and explained where the $n!$ cancels (which seems to have confused the above editor).

Thanks again – this makes it really slick.

—Nils von Barth (nbarth) (talk) 13:21, 17 August 2012 (UTC)

In addition to the elegant proof using the Gamma function, I’ve added back the “integration by parts” proof (in this edit), after the main proof, as an additional proof, primarily because this is how Bernoulli computed it initially (hence of historical interest), but also because it is straightforward approach, hence easier for some readers to follow and of educational use. Given a product integrand, integration by parts is the obvious attack, and in fact works. Recognizing the Gamma function requires additional background and insight, so it can feel rather deus ex machina, but I’ve noted that these are equivalent approaches (esp. b/c, as you note, one computes the Euler identity in the same way, by parts). Hope this is clear, and please feel free to continue to improve it!

—Nils von Barth (nbarth) (talk) 14:24, 17 August 2012 (UTC)

Good job Nils! At the moment, I do not see how to improve it ;) --pma 15:43, 22 August 2012 (UTC)

Actually, a (minor) objection is that the current version of the first proof seems to rely on the theory of the Gamma function. There is no need of the theory of the Gamma function, of course; the point of the computation was just to change variable in order to reach a well-known integral identity, namely $\int _{0}^{+\infty }x^{n}e^{-x}dx=n!$, which is already written in other wiki-articles and already known to many readers (also at Bernoulli's times: so that the parenthesis "the Gamma function (which was not yet known)" could be a bit misleading). I'd try to minimize the quotes to the Gamma function, in order to show the computaton elementary as it is. --pma 23:15, 19 September 2012 (UTC)

I don't know if it's just me, but I've tried recomputing this proof, and every time I try, your result looks like it forgot that

$dx=\exp \left({\frac {-u}{n+1}}\right)\left({\frac {-1}{n+1}}\right)\;du$.

That is, you seem to be missing a negative. Thus, when I substitute in that and

$x=\exp \left({\frac {-u}{n+1}}\right)$,

I end up with the expression

$\int _{0}^{1}x^{n}(\log x)^{n}\;dx=\int _{0}^{\infty }\exp \left({\frac {-un}{n+1}}\right)\left({\frac {-u}{n+1}}\right)^{n}\exp \left({\frac {-u}{n+1}}\right){\frac {-1}{n+1}}\;du$

= $\int _{0}^{\infty }\exp \left({\frac {-u(n+1)}{n+1}}\right)(-1)^{n}u^{n}{\frac {-1}{(n+1)^{n+1}}}\;du$

= $(-1)^{n+1}(n+1)^{-(n+1)}\int _{0}^{\infty }e^{-u}u^{n}\;du$,

which would yield the final result of

$\int _{0}^{1}x^{x}\;dx=\sum _{n=1}^{\infty }(-n)^{-n}$,

which differs from the proven result.

—Matt (JVz) (talk) 18:55, 22 September 2012 (UTC)

After the substitution you should probably get $(-1)^{n+1}(n+1)^{-(n+1)}{\mathbf {\int } _{\infty }^{0}}e^{-u}u^{n}\;du$ instead, and then use $\int _{\infty }^{0}=-\int _{0}^{\infty }$. pma 13:56, 25 March 2015 (UTC)

Freshman's dream in finite fields

The current version states that the freshman's dream is "incorrect unless one is working over a field or unital commutative ring of prime characteristic n or a factor of n. The correct result is given by the binomial theorem." I think this is false for the nonprime case (e.g., the identity does not hold for $n=6$ on a field of characteristic 2 or 3). ~[user:Fph] — Preceding unsigned comment added by 93.51.70.163 (talk) 20:04, 23 March 2015 (UTC)

You're so right, dude, we should seriously fix this — Preceding unsigned comment added by 160.39.202.48 (talk) 03:11, 31 March 2017 (UTC)

Generalization

$\sum _{n=k+1}^{\infty }n^{k-n}=\int _{0}^{1}t^{k-t}dt$

More at vixra.org/abs/1507.0051. To add this to the article is a good idea — Preceding unsigned comment added by 83.237.214.103 (talk) 09:19, 20 July 2015 (UTC)

This is a trivial statement which is definitely not notable.--94.224.66.163 (talk) 13:10, 8 November 2015 (UTC)

Rational or Irrational

I wanted to add Sophomore's dream to Template:Irrational number but is there a proof it is irrational?WillemienH (talk) 12:15, 3 July 2016 (UTC)

@WillemienH: Yes it is irrational as any convergent series $\sum _{n=1}^{\infty }{\frac {1}{\prod _{j=1}^{n}b_{j}}}$ such that any prime $p$ divides one of the $b_{j}\in \mathbb {N}$ — Preceding unsigned comment added by 78.196.93.135 (talk • contribs)

@78.196.93.135:, this series is not of the form you describe. --JBL (talk) 00:20, 27 June 2017 (UTC)

Algebraic or transcendental number

If this number is irrational, then how could we proof if this number is Algebraic or transcendental? — Preceding unsigned comment added by 190.113.30.229 (talk) 20:08, 24 January 2017 (UTC)