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  1. #1 just for fun... 
    Moderator Moderator AlexP's Avatar
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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+e<sup>b</sup>+e<sup>2b</sup>+e<sup>3b</sup>+...=9


    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  3. #2  
    Forum Professor serpicojr's Avatar
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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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  4. #3  
    Moderator Moderator AlexP's Avatar
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    Would anyone like to give it a try?
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  5. #4  
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    I’ll post the answer and leave the working to others who are still interested in giving it a try. 8)

    b = ln(8*⁄*9) = −ln(1.125)
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  6. #5  
    Forum Ph.D. william's Avatar
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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
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
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  7. #6  
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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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  8. #7  
    Forum Ph.D. william's Avatar
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    Quote Originally Posted by JaneBennet
    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
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
    ~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
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