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Math Help - Comparison Test Problems

  1. #1
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    Comparison Test Problems

    Hello All: I am discovering I am really bad at these comparison test problems -- of the 15 homework assignments, I have 4/5 that I can't do: please help me! While I don't want to beg for the answers, it would be v. helpful if someone could give me advice in setting these problems up/tips on completing them:

    #20: Sum (n=1 to Infinity) of (n + 4^n)/(n + 6^n)

    I get really confused in the whole "an/bn" business!

    #26: Sum (n=1 to Infinity) of n + 5/((n^7 + n^2)^(1/3))

    #28: Sum (n=1 to Infinity) of (e^(1/n))/n

    #34: Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

    Sum (n=1 to Infinity) of (sin^2(n))/(n^3)

    I said that (sin^2(n))/(n^3) is less than or equal to 1/(n^3), Rn is less than or equal to Tn is less than or equal to 1/2n^2, n = 10, R10 is less than or equal to .005, is this correct? How do I approximate the sum of the series?

    Eternal gratitude to whomever can assist with these problems! I really have been struggling with the comparison test -- everything else this chapter on series hasn't been too hard. Any advice is desperately and gratefully appreciated!!!!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Sprintz View Post
    Hello All: I am discovering I am really bad at these comparison test problems -- of the 15 homework assignments, I have 4/5 that I can't do: please help me! While I don't want to beg for the answers, it would be v. helpful if someone could give me advice in setting these problems up/tips on completing them:

    #20: Sum (n=1 to Infinity) of (n + 4^n)/(n + 6^n)

    I get really confused in the whole "an/bn" business!
    \frac{n+4^n}{n+6^n}<\frac{n+4^n}{n+6^n}

    and for large n: n<4^n, so:

    \frac{n+4^n}{n+6^n}<\frac{n+4^n}{6^n}<\frac{2\time  s 4^n}{6^n}=2\left(\frac{1}{3}\right)^n

    CB
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  3. #3
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    Quote Originally Posted by Sprintz View Post
    Hello All: I am discovering I am really bad at these comparison test problems -- of the 15 homework assignments, I have 4/5 that I can't do: please help me! While I don't want to beg for the answers, it would be v. helpful if someone could give me advice in setting these problems up/tips on completing them:

    [snip]

    #28: Sum (n=1 to Infinity) of (e^(1/n))/n

    [snip]
    Substitute the series expansion e^{1/n} = 1 + \frac{1}{n} + \frac{1}{2 n^2} + .... .

    Then you get \sum_{n=1}^{+\infty} \frac{1}{n} + \sum_{n=1}^{+\infty} \frac{1}{n^2} + \sum_{n=1}^{+\infty} \frac{1}{2n^3} + ....

    Now look at the first sum.


    Edit: Oops, I just noticed the title of your post. OK, note that \frac{e^{1/n}}{n} > \frac{1}{n} ....
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