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Pre Calc

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Write as a series and express as a rational number:

1. 5.36363636....

2. 0.0123123....




Use this series and find S1,S2,S3,S4,Sn, and lim Sn.

1/1*3 + 1/3*5 + 1/5*7+...+ 1/(2n-1)(2n+1)

  • Pre Calc -

    It works easier to have one type problem per question, as you will see here.
    On the first, try something like multiping by a power of ten that will align digits. For instance, in the first, multiply by 100
    536.36363636 . Subtract the original number
    536.3636-5.363636= 531
    Now divide by 99 (think out why)
    531/99 is your fraction

  • Pre Calc -

    I would split it this way:

    5.36363636....
    = 5 + (.36 + .0036 + .000036 + ...}

    so for the bracket part, a=.36, r = .01

    remember S = a/(1-r)

    = .36/(1-.01) = .36/.99 = 36/99 = 4/11

    then 5.36363636.... = 5 4/11 or 59/11

    do the rest the same way

  • Pre Calc -

    1. 5.36363636.... = 5 + 0.36(1 + 10^-2 + 10^-4 + ..)
    = 5 + 0.36[1/(1-.01)]
    = 5 + 0.36 * 100/99
    = 5 + 36/99

    2. Do it the same way

    3. If you are dealing with sums, the first term is
    S1 = 1/3
    and the second term is
    S2 = 1/3 + 1/15 = 6/15
    Do the others and see what Sn and the limit are

  • Pre Calc -

    for your last question...
    1/1*3 + 1/3*5 + 1/5*7+...+ 1/(2n-1)(2n+1)

    S1 = 1/3
    S2 = 1/3 + 1/15 = 6/15 = 2/5
    S3 = 2/5 + 1/35 = 15/35 = 3/7

    do you see a pattern?
    so what is Sn ?

    An interesting question now would be,
    Prove that your answer to the above is correct by induction.

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