Emerald Stoker Posted March 1, 2019 Share Posted March 1, 2019 The Kid (who is much better at math than I am) is working on a chapter about the mean-value theorem (which is otherwise going pretty well so far), but is stuck on the problem below; we spent most of yesterday afternoon trying to wrestle with it without much luck. It sort of "feels" as though the proof should be simple and interesting, but so far it's just a head-scratcher! (It's from Sherman Stein's Calculus in the First Three Dimensions.) Using the inequality ex > 1 + x for x > zero, prove this theorem: If u1, u2, u3,…is a sequence of positive numbers such that the sequence of sums u, u1 + u2, u1 +u2 + u3, … converges, then the sequence of products (1 + u1), (1 + u1)(1 + u2), (1 + u1)(1+u2)(1+u3),… also converges. Then prove the converse. Would any of you very kind math people be able to give us a push in the right direction? We'd be so very, very appreciative! Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted March 1, 2019 Share Posted March 1, 2019 Think about it this way: Since the ui >0, you know also that ui + 1 >0. You need to show that the infinite product (u1 +1 )*(u2 + 1)* (u3 + 1).... converges. But by the given inequality at the top of the problem, you know that each ui + 1 < exp(ui), so the product (u1 +1)*(u2 +1)*(u3 +1)...< exp(u1)*exp(u2)*exp(u3)....=exp (u1 + u2 + u3 + ...) by exponent rules. Since the infinite series u1 + u2 + u3 +.... converges (given), it has a sum S which is a positive real number. Then (u1 + 1)*(u2 + 1)*(u3 + 1)...< exp(S), showing that the infinite product converges. Hope that helps 🙂 I'll leave the proof of the converse to your son. 1 Quote Link to comment Share on other sites More sharing options...
Emerald Stoker Posted March 1, 2019 Author Share Posted March 1, 2019 Thank you so much, Kathy! That makes excellent sense. (You should have seen the smile on his face when he read your post--he's run off to prove the converse now!) Thank you for your generous help--we appreciate it. 1 Quote Link to comment Share on other sites More sharing options...
Emerald Stoker Posted March 2, 2019 Author Share Posted March 2, 2019 Thanks, square_25! There are indeed some interesting problems in this book. We used his geometry and alg2/trig books, too--(those were both co-written by Chakerian and Crabill)--so we picked this calculus book based on warm feelings for the first two! And also because there is a whole chapter about applications of calculus to traffic (this is my child who's fascinated with urban planning)! I'm just hanging on by my fingernails, though, now that we've gotten to calculus, which I never studied myself--good thing The Kid loves math, and can usually explain it to me! And thank goodness for the hive when he can't....thanks again to you both. Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted March 2, 2019 Share Posted March 2, 2019 I agree with square_25. That sounds like a really cool calc book. I own about a gazillion math books, but i’m thinking that I need another one now, hmmm! Good luck and wishes for happy math adventures to your son 😀 1 Quote Link to comment Share on other sites More sharing options...
Emerald Stoker Posted March 2, 2019 Author Share Posted March 2, 2019 Thanks, Kathy! It has indeed been such a lovely adventure--I have been feeling very nostalgic lately about all of the fun stuff we used to do--Miquon, and the Anno books, and Malba Tahan, and the Thos. Crowell Young Math Books, lots of picture books about math, and the Nuffield green/red/purple books, Don Cohen's calculus for six-year-olds (or whatever that cute little spiral-bound book was called), lots of recreational problem-solving books--math was glorious fun! It still is for him, which makes me really glad (not so fun for me now that I feel so out of my depth, but every now and then I still get a few glimmers and feel very happy!). You have been a big help along the way, too--I carefully noted your recommendations for things as my kids grew--lovely Jacobs for algebra and the Shanks/Brumfiel/Fleenor/Eicholz precalc book were big hits with him, and I found them in your posts first! The Stein calculus book is published by Dover, so nice and cheap! And there are some instructor's materials free on Dover's website (not that you would need them!). We have found some gems on their site--we got some books Kiana recommended that my son loved (about topology and graph theory and game theory), and years ago I got him the huge Averbach/Chein recreational math book--that was perfect when he was about eight or so--he'd finished the AoPS number theory book at that point and was hungry for more mathy fun. 1 Quote Link to comment Share on other sites More sharing options...
Emerald Stoker Posted March 2, 2019 Author Share Posted March 2, 2019 Oh, age 6 is the best! I loved it. So many things to explore, so much fun to try to figure out how little brains work! I kept big baskets of math picture books around--and we'd also just get out pencils and paper and make up puzzles for each other. It was so joyous! What I found so interesting was the sophistication of the concepts that we could try--little kids don't know things are supposed to be hard, I think! So we talked a lot about number bases (we had fun books called The History of Counting and How to Count Like a Martian, I remember) and other fun things--lots of geometry-type things, too. I wish I could remember more, or had kept a journal. I do remember his favourite math thing when he was five--taking out a square piece of graph paper and colouring half of it, then colouring half of the remaining half in a different colour, and so on and so on, until he was left with the tiniest little bit in the middle--that was good for hours of contemplation and adding up fractions...Also he made lots of graphs of data about his books when he was six or so (especially the Hardy Boys since there are so many of them): frequency of repeated words in the titles or scenes in the illustrations...lots and lots of graphs. Oh, and also lots of fun--we'd go out to the driveway and I'd set up equations with pinecones and seashells in place of variables...or we'd make tower of Hanoi puzzles with different sizes of leaves from around the yard. It was so so so fun. I'm glad you're hanging out on this board! 1 Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted March 2, 2019 Share Posted March 2, 2019 Oh cool, thanks for the info! When I get home from traveling, I’m going to dig out my Dover catalog 🙂 i *still* have my whole collection of fun math books on the shelf that I started accumulating when my boy was little (including that cute Don Cohen spiral book)...and he’ll be 31 next week! And I may have a couple from when I was a little girl, too (but not telling my age, lol). Isn’t the Shanks et al precalc great? I even brought my copy on this trip...It’s been helpful with one of my better tutoring kids this year. 🙂 1 Quote Link to comment Share on other sites More sharing options...
Emerald Stoker Posted March 2, 2019 Author Share Posted March 2, 2019 Yes, I liked that book a lot--really good explanations, and some very challenging problems. I hope you're enjoying your trip! Quote Link to comment Share on other sites More sharing options...
Emerald Stoker Posted March 2, 2019 Author Share Posted March 2, 2019 (edited) I'm trying to remember...all of the Mitsumasa Anno books were wonderful (Anno's Magic Seeds, Anno's Mysterious Multiplying Jar, Anno's Math Games). Kathryn Lasky, The Librarian Who Measured the Earth. Denise Schmandt-Besserat, The History of Counting. Glory St. John, How to Count Like a Martian. Demi, One Grain of Rice. Marilyn Burns, The I Hate Mathematics Book (which I kept in a plain brown wrapper so he couldn't see the title!!). Hans Magnus Enzensberger, The Number Devil. Theoni Pappas, the Penrose books. Puzzlegrams by Pentagram Press. David Schwartz, G is for Googol. The Thomas Y Crowell Young Math series (there are about twenty-odd of them, I think--mostly out of print). Catherine Sheldrick Ross, Circles, Squares, and Triangles (three separate books). Charles Townsend, Merlin's Big Book of Puzzles, Games and Magic. There were more, but I can't remember any more, sadly. The Nuffield books were fun (though not picture books): https://www.stem.org.uk/elibrary/resource/28047 The Living Math website always had a lot of good suggestions, too: https://www.livingmath.net/reader-index Hope you'll find something you'll like! ETA: Ivar Ekeland, The Cat in Numberland!! Awesome book. Edited March 2, 2019 by Emerald Stoker 1 Quote Link to comment Share on other sites More sharing options...
Emerald Stoker Posted March 2, 2019 Author Share Posted March 2, 2019 Humour might be good? The Murderous Maths series is a bit of a hoot. Quote Link to comment Share on other sites More sharing options...
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