From mboxrd@z Thu Jan 1 00:00:00 1970 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Newsgroups: gmane.emacs.help Subject: Re: The next exercise Date: Sat, 10 Dec 2022 07:18:07 +0100 Message-ID: References: <87tu24tmv6.fsf@web.de> <87y1rgs472.fsf@web.de> Mime-Version: 1.0 Content-Type: multipart/signed; micalg=pgp-sha1; protocol="application/pgp-signature"; boundary="dSq7JKc5by5CYF4K" Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="3721"; mail-complaints-to="usenet@ciao.gmane.io" To: help-gnu-emacs@gnu.org Original-X-From: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane-mx.org@gnu.org Sat Dec 10 07:18:42 2022 Return-path: Envelope-to: geh-help-gnu-emacs@m.gmane-mx.org Original-Received: from lists.gnu.org ([209.51.188.17]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1p3tC6-0000m2-Ob for geh-help-gnu-emacs@m.gmane-mx.org; Sat, 10 Dec 2022 07:18:42 +0100 Original-Received: from localhost ([::1] helo=lists1p.gnu.org) by lists.gnu.org with esmtp (Exim 4.90_1) (envelope-from ) id 1p3tBi-0000uO-6K; Sat, 10 Dec 2022 01:18:18 -0500 Original-Received: from eggs.gnu.org ([2001:470:142:3::10]) by lists.gnu.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.90_1) (envelope-from ) id 1p3tBe-0000uF-TG for help-gnu-emacs@gnu.org; Sat, 10 Dec 2022 01:18:15 -0500 Original-Received: from mail.tuxteam.de ([5.199.139.25]) by eggs.gnu.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.90_1) (envelope-from ) id 1p3tBc-0004Qm-N3 for help-gnu-emacs@gnu.org; Sat, 10 Dec 2022 01:18:14 -0500 DKIM-Signature: v=1; a=rsa-sha256; q=dns/txt; c=relaxed/relaxed; d=tuxteam.de; s=mail; h=From:In-Reply-To:Content-Type:MIME-Version:References:Message-ID: Subject:To:Date:Sender:Reply-To:Cc:Content-Transfer-Encoding:Content-ID: Content-Description:Resent-Date:Resent-From:Resent-Sender:Resent-To:Resent-Cc :Resent-Message-ID:List-Id:List-Help:List-Unsubscribe:List-Subscribe: List-Post:List-Owner:List-Archive; bh=qi7cf/o9QH2ogqs8+ZHJwfvT6oOkiSt7mBYB67BN0vs=; b=od48Qu12iB8oOFDLct9VtF6USl HhStPyKsVNiEFO7CeRd0prnKRGRqe65tfh/T0HsPo+DM1gND9Lvp1ysuJ0m2CbzUkZXAKHwrLlOYD 2ti1kvWdywRFgMPFFwvGazDESqwWy2BLX65I47EUVgxAh97fCmmSgXSdp8v3kaLMHl0MuJQwtGJSK KCNaon2HRThsQZ9MBB7dLd/ddy8huLlclugQ9fUxAycRnkCY8uRmmEU3SmpfZdNmIQu+SuVApDUc8 9MZg8NpASXCEttwRYGMZmyIMv0AjXq1Ky1N+49jSTKpgSlOUahL+MTC15IzvzaWFczUMCdOgZ1MQ+ IAg5dXvA==; Original-Received: from tomas by mail.tuxteam.de with local (Exim 4.94.2) (envelope-from ) id 1p3tBX-00036u-Hz for help-gnu-emacs@gnu.org; Sat, 10 Dec 2022 07:18:07 +0100 Content-Disposition: inline In-Reply-To: <87y1rgs472.fsf@web.de> Received-SPF: pass client-ip=5.199.139.25; envelope-from=tomas@tuxteam.de; helo=mail.tuxteam.de X-Spam_score_int: -20 X-Spam_score: -2.1 X-Spam_bar: -- X-Spam_report: (-2.1 / 5.0 requ) BAYES_00=-1.9, DKIM_SIGNED=0.1, DKIM_VALID=-0.1, DKIM_VALID_AU=-0.1, DKIM_VALID_EF=-0.1, SPF_HELO_NONE=0.001, SPF_PASS=-0.001 autolearn=ham autolearn_force=no X-Spam_action: no action X-BeenThere: help-gnu-emacs@gnu.org X-Mailman-Version: 2.1.29 Precedence: list List-Id: Users list for the GNU Emacs text editor List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Errors-To: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane-mx.org@gnu.org Original-Sender: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane-mx.org@gnu.org Xref: news.gmane.io gmane.emacs.help:141545 Archived-At: --dSq7JKc5by5CYF4K Content-Type: text/plain; charset=utf-8 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable On Fri, Dec 09, 2022 at 09:38:25PM +0100, Michael Heerdegen wrote: > writes: >=20 > > My choice was school maths [...] [...] > Or you remember how e can be introduced: with >=20 > / 1\ n / 1\ n+1 > a_n :=3D |1 + -| and b_n :=3D |1 + -| =20 > \ n/ \ n/ =20 FWIW, I started with the binomial formula, (1000 + 1)^1000 =3D 1000^1000 + (n-over-1) * 1000^999 + ... Now n-over-1 is 1000, so we already have a 2 at the left (more to come, of course :) Next (n-over-2) is 1000*999/1*2 (so roughly 1000*1000/2, that adds 0.5 to the most significant, that makes 25...; Next 1000^3/6, so we are at 2666... The denominator grows quickly (a tad more than an exponential), so things are bound to converge quickly; besides, I over-estimated things by setting 99x to 1000. That already started looking a lot like e. Then I remembered that limit I quoted from you above. Then I asked Guile and Emacs. Cheers --=20 t --dSq7JKc5by5CYF4K Content-Type: application/pgp-signature; name="signature.asc" -----BEGIN PGP SIGNATURE----- iF0EABECAB0WIQRp53liolZD6iXhAoIFyCz1etHaRgUCY5QkmAAKCRAFyCz1etHa RjE/AJ4lYgGJd1nVx2OCQlFOPyK9mAf3UACaAygjcnqwgAtWd3ORYXZDIFakAi4= =cD/z -----END PGP SIGNATURE----- --dSq7JKc5by5CYF4K--