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| Chemboy |
 Posted: Tue May 06, 2008 8:45 pm Post subject: just for fun... |
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Just for fun... It was a bonus question on a calc test and I liked it. Maybe those of you for whom it's likely very easy could hold off for a little while and see how other people do with it.
Find the value of b for which 1+eb+e2b+e3b+...=9
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| serpicojr |
| Posted: Tue May 06, 2008 8:55 pm Post subject: |
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| An excellent question indeed! I like how it turns what would be a fairly routine problem in, say, second semester calculus on its head. |
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| Chemboy |
| Posted: Thu May 08, 2008 10:54 am Post subject: |
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Would anyone like to give it a try?
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| JaneBennet |
| Posted: Thu May 08, 2008 12:36 pm Post subject: |
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I’ll post the answer and leave the working to others who are still interested in giving it a try. 
b = ln(8 ⁄ 9) = −ln(1.125)
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| william |
| Posted: Thu May 08, 2008 1:50 pm Post subject: |
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Crikey this took me a lot of steps. There must be an easier way than what I did.
I refuse to write out my solution without latex, but here is the steps in words:
1. Integrate the original equation,
2. notice the result has the form ln(1+e^b) + stuff,
3. differentiate that,
4. notice a geometric series in the result,
5. here you get "junk = 10",
6. massage "junk" by factoring, canceling, expanding, etc.,
7. you should end up with (e^b+1)(-8+9e^b)=0,
8. BAM! solve for b.
So my solution involved integration, differentiation, use of the series ln(1+x), use of a geometric series, expanding, and factoring.
Cheers,
william
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| JaneBennet |
| Posted: Thu May 08, 2008 4:12 pm Post subject: |
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Well, actually, there is no need for integration or differentiation. The whole thing is an infinite geometric series. 
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| william |
| Posted: Fri May 09, 2008 10:35 am Post subject: |
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| JaneBennet wrote: |
| Well, actually, there is no need for integration or differentiation. The whole thing is an infinite geometric series. |
Shiiiiit! I'll be damned. That makes it a piece of cake! A 10-second problem. 
Thanks,
william
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~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
About my avatar: This is a smoothed particle hydrodynamics (SPH) simulation of the merger of two galaxies. The code was written by Volker Springel of the Max Planck Institute for Astrophysics at Garching Germany. This simulation uses 20,000 disk particles (stars) and 40,000 halo particles (dark matter) per galaxy. The three views are, from left to right, the x-y plane, x-z plane, and y-z plane. |
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