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Math Help - few simple limit problems :D

  1. #1
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    few simple limit problems :D

    hello,
    I'm having problems with few limits... I have done them but my solution isn't the same as in the book... so can anyone check them or point me what do I do wrong...

    (sorry but LATEX don't work I'll fix it when it does)

    first problem ::

    \lim_{n\to \infty} \frac{1-2+3- ... + (-1)^(n-1) n}{ n}

    I try using Stolz th but come to way wron solution...

    so I look at the 1-2+3- .... as two series ... one with odd and one with even numbers like this :

    \sum_{n=0} ^{\infty} (2n+1) - \sum_{n=0} ^{\infty} 2n

    so i find nth partial sum of each of them and got this

    (n+1)^2 (from the odd ) and n(n+1) (from the even numbers)

    so combined those two i get to nth partial sum be just (n+1 )

    after that limit looks like :

    \lim_{n\to \infty} \frac {n+1}{n}

    and solution is 1 but that's not true


    second problem is :
    show that
    \lim_{n\to \infty} \sqrt[n] {a_1 * a_2 * .... a_n } = \lim_{n\to \infty} a_n

    hmmmm here I say that some A_n is A_n = \sqrt[n] {a_1 * a_2 * .... a_n }

    so now I do log

    \ln (A_n) = \frac{1}{n} * (\ln(a_1) + \ln(a_2) +.... +\ln(a_n)

    so i use Stolz th

    \lim_{n\to \infty} \frac {x_(n+1) - x_n}{y_{n+1}-y_n}

    and I get to be equal to \lim_{n\to \infty} a_(n+1) not a_n (since all another just substract.. )

    Third problem :

    is the same as second ... just with numbers like


    \lim_{n\to \infty} \sqrt[n] {(1+1/1)^1 * (1+1/2)^2 * .... (1+1/n)^n }

    and I get to result be "e" hope it's correct



    thank you all very very much
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  2. #2
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    In the first question I like your idea of evaluating it as two separate series, but they are arithmetic series.

    For the odd terms 1 + 3 + 5 + ... there would be n/2 terms, and for -(2 + 4 + 6 + ...) there would be n/2 terms. Use S = N[2a + (N - 1)d]/2 for each.
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  3. #3
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    Thank you

    perhaps it's stupid question but I'm not familiar with what do represent there "2a" and "d" what you have wrote there


    Edit :

    I have found in one of the books that Stolz th goes like this ...

    \lim_{n\to \infty} \frac{x_n - x_{n-1}}{y_n - y_{n-1}} ... and like that i get second ok does it matter which do I use ? or are they the same
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  4. #4
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    a is the first term and d is the common difference.
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  5. #5
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    hmmm I think I may have done another stupid mistake

    for (1+3+5+..... ) S_(odd)= N[2*1 + (N-1)*2] /2 = .... = N^2
    for -(2+4+6+.... ) S_(even) = N[2*2 + (N-1)*2]/2 = .... = N^2 +N

    so combined S_(odd) - S_(even) = -N

    \lim_{n\to \infty} \frac{-n}{n}

    and than I get limit to be -1 ?
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  6. #6
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    No, N and n are not the same thing. I've used N to represent the number of terms in the smaller series, while n represents the number of terms in the alternating series.

    So N = n/2.
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  7. #7
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    aaah I got it thank you very very much
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