tag:blogger.com,1999:blog-264226589944705290.post7856622831041493292..comments2023-11-05T03:45:25.001-08:00Comments on God Plays Dice: How could you guess the formula for the sum of the first n fifth powers?Michael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-264226589944705290.post-52736236303747231102008-12-05T15:12:00.000-08:002008-12-05T15:12:00.000-08:00A. Rex,of course you're right, but I'm not sure th...A. Rex,<BR/><BR/>of course you're right, but I'm not sure that the etymology would have been obvious to everybody, and it's possible that you were just commenting that the word "shadowy" seems somehow atypical in mathematical writing.Michael Lugohttps://www.blogger.com/profile/15671307315028242949noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-27007340347303178552008-12-05T14:44:00.000-08:002008-12-05T14:44:00.000-08:00Michael, I'm aware of course that "umbral" means "...Michael, I'm aware of course that "umbral" means "shad[ow]y". But I think there's a distinction between having a name and having a property, e.g. are perverse sheaves actually?<BR/><BR/>unapologetic, I meant to link to your post myself, but I couldn't remember where I had seen it. I searched for "Faulhaber's * Formula" on Google, but it didn't find it. PS. I think that, surprisingly, you're the first person to call me by the intended abbreviated form of my name.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-41950168949393423232008-12-05T05:43:00.000-08:002008-12-05T05:43:00.000-08:00Okay, I was trying not to self-promote too much, b...Okay, I was trying not to self-promote too much, but A. Rex has pushed me to link to <A HREF="http://unapologetic.wordpress.com/2007/11/14/faulhabers-fabulous-formula/" REL="nofollow">my own coverage</A>.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-78493913048130923382008-12-04T18:12:00.000-08:002008-12-04T18:12:00.000-08:00"umbral" means "shadowy"."umbral" <I>means</I> "shadowy".Michael Lugohttps://www.blogger.com/profile/15671307315028242949noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-48790306099185093182008-12-04T18:10:00.000-08:002008-12-04T18:10:00.000-08:00Faulhaber's fantastic formula!!In all seriousness,...<A HREF="http://en.wikipedia.org/wiki/Faulhaber%27s_formula" REL="nofollow">Faulhaber's fantastic formula</A>!!<BR/><BR/>In all seriousness, there are many ways to prove the existence of (or derive) the formula for the sums of powers, but the umbral calculus is amazing. I would encourage anyone who hasn't seen this approach before (which Wikipedia even describes as "shadowy") to read about it.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-10302410070165946682008-12-04T15:16:00.000-08:002008-12-04T15:16:00.000-08:00One easy way to remember the sum of k^th powers.in...One easy way to remember the sum of k^th powers.<BR/><BR/>integral(s_k dn)+constant=s_{k+1}ashwin kumar b vhttps://www.blogger.com/profile/07777870747608944808noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-2730840036001105202008-12-04T13:56:00.000-08:002008-12-04T13:56:00.000-08:00Or you could use the calculus of fininte differenc...Or you could use the calculus of fininte differences in which case you'd get the formula and the proof that it exists for all n at the same time.CarlBrannenhttps://www.blogger.com/profile/17180079098492232258noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-83641698952446619672008-12-04T13:45:00.000-08:002008-12-04T13:45:00.000-08:00I would have first guessed based on the pattern th...I would have first guessed based on the pattern that it would be a 6th-degree polynomial, then computed the polynomial from the first 6 elements, then checked that it fit the rest of the sequence.<BR/><BR/>(Not this is a mathematically deep approach, but it's what I first instinct would have been if you'd asked me to figure out the sequence).Unknownhttps://www.blogger.com/profile/00722476612538640230noreply@blogger.com