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Math Help - Sequences

  1. #1
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    Sequences

    A_n = (1 + (3/n) )^4n

    I don't know hot to find if this converges. I feel like I should break up the exponent, but I don't know how that would help.
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  2. #2
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    This uses a couple of neat tricks.

    Let this limit equal A. Then take the natural log of both sides and you get:

    \ln(A)=4n\ln(1+\frac{3}{n})

    \frac{\ln(1+\frac{3}{n})}{(4n)^{-1}}=\ln(A)

    Now if you try to evaluate this limit for n -> infinity, you get an indeterminate form, so you can use L'Hopitals rule.

    \frac{(\frac{n}{n+3}) (\frac{-3}{n^2})}{\frac{-4}{n^2}}=\ln(A)

    The right hand side stays the same because after we apply the rule, nothing changes about the limit.

    Now it's just simplifying and you should get it down to:

    \frac{3n}{4n+12}=\ln(A)

    Raise both sides to the power of e and you get A=e^{\frac{3}{4}}
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  3. #3
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    That is pretty neat, but is there any other way to do that? Since we haven't done anything like that yet in class, I feel like there should be a more obvious way...
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  4. #4
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    I never learned the part you're asking me to remember. Can you show me why that is true?
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  5. #5
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    Hello, veronicak5678!


    We're expected to know this fact:

    . . \lim_{z\to\infty}\left(1 + \tfrac{1}{z}\right)^z \;=\;e



    A_n \:= \:\left(1 + \tfrac{3}{n}\right)^{4n}
    We want: . \lim_{n\to\infty}\left(1 + \tfrac{3}{n}\right)^{4n}


    Multiply the exponent by \tfrac{3}{3}

    . . \left(1 + \tfrac{3}{n}\right)^{4n\cdot\frac{3}{3}} \;=\;\left(1 + \tfrac{3}{n}\right)^{\frac{n}{3}\cdot12} \;=\; \bigg[(1 + \tfrac{3}{n})^{\frac{n}{3}}\bigg]^{12} \;= \bigg[\left(1 + \frac{1}{\frac{n}{3}}\right)^{\frac{n}{3}}\bigg]^{12}


    \text{Let }\:z \,=\,\tfrac{n}{3}

    . . and we have: . \lim_{z\to\infty}\bigg[\left(1 + \frac{1}{z}\right)\bigg]^{12} \;=\;\bigg[\lim_{z\to\infty}\left(1 + \tfrac{1}{z}\right)^z\bigg]^{12} \;=\;e^{12}

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  6. #6
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    I see! I'm beginning to think this was in the part of the lecture my teacher never got to before time ran out. Thanks so much for your help!
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  7. #7
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    I made a mistake obviously somewhere. My apologies. Soroban was there to save the day as usual.
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