tag:blogger.com,1999:blog-7423697418269914273.post3582130643747518202..comments2016-10-07T18:13:58.592-07:00Comments on Paul Liu: The Kempner Series - A modified harmonic seriesPaul Liuhttp://www.blogger.com/profile/16809371907394009052noreply@blogger.comBlogger18125tag:blogger.com,1999:blog-7423697418269914273.post-8912011741680968112014-03-06T16:55:35.980-08:002014-03-06T16:55:35.980-08:00youre a great help to current Science One-ers :)youre a great help to current Science One-ers :)celine shttps://www.blogger.com/profile/09685027841701524337noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-51644634311967696062013-10-25T23:21:10.815-07:002013-10-25T23:21:10.815-07:00Very nicely done , will help me a lot for my upcom...Very nicely done , will help me a lot for my upcoming seminar! Thanks Neeraj Bhauryalhttps://www.blogger.com/profile/00228943383608556339noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-7849050605113668002012-05-11T12:46:57.200-07:002012-05-11T12:46:57.200-07:00Yes! The powers of two thing is actually pretty us...Yes! The powers of two thing is actually pretty useful in testing for convergence or divergence. An elementary proof of the harmonic series divergence can be found by combining terms in groups of powers of two. <br /><br />What I mean is: 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... = 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ... >= 1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ... >= 1 + 1/2 + 1/2 + 1/2 + ...<br />which diverges quite obviously.<br /><br />Note that the above is just a special case of the Cauchy Condensation Test, which makes even better use of powers of two.<br /><br />Cheers,<br />PaulPaul Liuhttps://www.blogger.com/profile/16809371907394009052noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-59042594958925193202012-05-11T10:09:10.155-07:002012-05-11T10:09:10.155-07:00This was a pretty interesting read, I'd never ...This was a pretty interesting read, I'd never heard of this before. Another obvious way to make the series converge is to raise the denominator of each term of the harmonic series to a power greater than or equal to 2.Justin Archerhttps://www.blogger.com/profile/02417547954183994873noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-15229717149275590342011-10-05T13:23:52.036-07:002011-10-05T13:23:52.036-07:00@Anonymous
That was a pretty interesting read! Th...@<a href="#c2209092788012662545" rel="nofollow">Anonymous</a><br /><br />That was a pretty interesting read! The code they provide allow you to find the sum of any Kempner-like series you wish. I am particularly curious about one thing though. All of these sums are particularly not likely to be rational. Could we find a Kempner-like series that converges to a rational sum?Paul Liuhttps://www.blogger.com/profile/16809371907394009052noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-22090927880126625452011-10-05T11:24:58.221-07:002011-10-05T11:24:58.221-07:00Here's another variation on the problem, from
...Here's another variation on the problem, from<br />http://front.math.ucdavis.edu/0806.4410<br /><br />The sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428708074784831968Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-44676944875542871152011-08-29T19:19:15.582-07:002011-08-29T19:19:15.582-07:00@Thomas Schmelzer
Haha yes! He probably was at th...@<a href="#c5177688879828111227" rel="nofollow">Thomas Schmelzer</a><br /><br />Haha yes! He probably was at that seminar. I know he was at Oxford just a couple years ago. <br />Thanks for the book recommendation and thank you very much for the encouragement! It's kind words from people such as yourself that inspire me to write these expositions on math.<br /><br />PaulPaul Liuhttps://www.blogger.com/profile/16809371907394009052noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-51776888798281112272011-08-29T15:49:24.780-07:002011-08-29T15:49:24.780-07:00I am delighted you find this interesting :-) Your ...I am delighted you find this interesting :-) Your professor isn't that old ;-) I think he might have been in the seminar I gave at Oxford discussing this problem. I highly recommend you get a copy of Alex's Adventures in Numberland by Alex Bellos. He is discussing the same problem (and many other even more interesting ones). Your plots are great! <br /><br />Enjoy your studies<br />ThomasThomas Schmelzernoreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-72114259573671313292011-07-10T01:05:16.032-07:002011-07-10T01:05:16.032-07:00@Anonymous
Hmm. I thought it would make my page to...@<a href="#c6106626770225910375" rel="nofollow">Anonymous</a><br />Hmm. I thought it would make my page too long to load due to the amount of LaTeX but I think I will increase the number of posts per page now.<br /><br />Thanks, anonymous helper!Paul Liuhttps://www.blogger.com/profile/16809371907394009052noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-61066267702259103752011-07-09T20:37:32.684-07:002011-07-09T20:37:32.684-07:00now you ask for suggestions, what about increase t...now you ask for suggestions, what about increase the number of post per page?<br /><br />at the moment it's not a big deal, but in some time you will have (i hope) a decent amount of posts, and it's quite slow trying for anyone who tries to find something they want to readAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-89737790865793334752011-07-04T02:37:22.937-07:002011-07-04T02:37:22.937-07:00@Chao Xu
Thanks! I'll take a look in the morni...@<a href="#c4600316064736048967" rel="nofollow">Chao Xu</a><br />Thanks! I'll take a look in the morning. It'll give me a good chance to take the rust off my chinese reading skills too.Paul Liuhttps://www.blogger.com/profile/16809371907394009052noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-46003160647360489672011-07-04T01:46:58.730-07:002011-07-04T01:46:58.730-07:00@Paul Liu
Here is the link to the paper
Of course...@<a href="#c2049272114613524954" rel="nofollow">Paul Liu</a><br /><br />Here is the link to the paper<br />Of course the journal isn't a well established one. It's just a journal for one university.<br />@Pual<br />http://wenku.baidu.com/view/1ab74fb169dc5022aaea004b.html<br />I assume you can read Chinese...<br /><br />It is very LOL worthy.<br /><br />Almost as funny as the medical doctor who rediscovered Riemann sum, published in a medical journal, and manage to get 70+ citations...Chao Xuhttp://chaoxuprime.comnoreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-20492721146135249542011-07-03T19:27:30.143-07:002011-07-03T19:27:30.143-07:00@Chao Xu
Ah. What journal is this? I might take a ...@<a href="#c8864448636505652145" rel="nofollow">Chao Xu</a><br />Ah. What journal is this? I might take a look if I have time. <br /><br />Also, sorry to digress on this reply but I'd like to explain to future readers as to why the modified harmonic series containing only terms with 9's is <b>not</b> convergent (as you've said).<br /><br />If the series containing only 9's (i.e. the part we removed from the harmonic series) is convergent, then that means the sum of all the terms we removed from the harmonic series is <b>finite</b>. However, if we only removed a finite portion from the infinite sum of the unmodified harmonic series, we could not have possibly made the series convergent as we proved. Hence, the series containing only terms with 9's is divergent and must sum to infinity.Paul Liuhttps://www.blogger.com/profile/16809371907394009052noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-88644486365056521452011-07-03T19:18:06.302-07:002011-07-03T19:18:06.302-07:00Very nice...
I checked out this post because I onc...Very nice...<br />I checked out this post because I once saw an article posted in a Chinese journal that proves the Harmonic series is both divergent and convergent. iirc, it uses the "remove all 9" argument too. But the author showed the sum of terms containing at least a 9 is convergent(which is horribly wrong...).Chao Xuhttp://chaoxuprime.com/noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-46506686279985486382011-07-03T15:41:26.010-07:002011-07-03T15:41:26.010-07:00@Anonymous
Thank you very much!
Suggestions on mak...@<a href="#c7097808221841172552" rel="nofollow">Anonymous</a><br />Thank you very much!<br />Suggestions on making the blog better are always welcomed (same to all the anons above)!Paul Liuhttps://www.blogger.com/profile/16809371907394009052noreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-70978082218411725522011-07-03T14:18:18.653-07:002011-07-03T14:18:18.653-07:00I feel very stupid now. Seems interesting.I feel very stupid now. Seems interesting.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-47471882166795848982011-07-03T10:02:45.010-07:002011-07-03T10:02:45.010-07:00Very nicely explained!Very nicely explained!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7423697418269914273.post-71932692824565677682011-07-03T06:45:42.927-07:002011-07-03T06:45:42.927-07:00This is quite interesting
Nice blog you got here ...This is quite interesting<br /><br />Nice blog you got here by the way<br /><br />keep it upAnonymousnoreply@blogger.com