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Math Help - Sum of the series

  1. #1
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    Sum of the series

    Write down in full the sum of each of the following series , and hence find its value .

    (1) \sum^{5}_{r=1}\frac{90}{r}

    (2) \sum^{8}_{r=1}\sin\frac{r\pi}{3}

    (3) \sum^{5}_{r=0}(-1)^r(1+2^{r+1})

    My thought process :

    (1) 90\sum^{5}_{r=1}\frac{1}{r}
    =90\frac{2}{n(n+1)}
    =90\frac{2}{5(6)}
    =6 ---- my ans is wrong .

    I have no idea how to do the rest .
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  2. #2
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    well, if you cannot think of the general of the sum, then use common sense. since the nth number is quite less, we can substitute the value one by one.
    for example, for first question, it will be
    90(1/1 + 1/2+ 1/3+ 1/4+ 1/5)
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  3. #3
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    Re :

    I am sorry for not including this in my previous post . How to apply the formulas to find the sum of these series instead of listing down all the terms and adding them up ?
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  4. #4
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    Quote Originally Posted by thereddevils View Post
    Write down in full the sum of each of the following series , and hence find its value .

    (1) \sum^{5}_{r=1}\frac{90}{r}

    (2) \sum^{8}_{r=1}\sin\frac{r\pi}{3}

    (3) \sum^{5}_{r=0}(-1)^r(1+2^{r+1})

    My thought process :

    (1) 90\sum^{5}_{r=1}\frac{1}{r}
    =90\frac{2}{n(n+1)}
    =90\frac{2}{5(6)}
    =6 ---- my ans is wrong .

    I have no idea how to do the rest .
    Of course your answer is wrong. \sum^{n}_{r=1}\frac{1}{r} \neq \frac{1}{\sum^{n}_{r=1} r}. eg. 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \neq \frac{1}{1 + 2 + 3 + 4 + 5} should be obvious.

    Just write each sum out in full and then add up using basic arithmetic.

    Quote Originally Posted by thereddevils View Post
    I am sorry for not including this in my previous post . How to apply the formulas to find the sum of these series instead of listing down all the terms and adding them up ?
    There is certainly no formula for the sum of reciprocals. The fact that there are only a few terms in each of your sums should be the clue that you write the sum out in full and then calculate.
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  5. #5
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    Thanks . Is it the same for (2) and (3) ?
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  6. #6
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    Quote Originally Posted by thereddevils View Post
    Thanks . Is it the same for (2) and (3) ?
    Yes.
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  7. #7
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    yea, the method is the same. for example,
    (2) would be:
    sin(pi/3)+sin(2pi+3)+...+sin(8pi/3)
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  8. #8
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    <br />
\sum^{8}_{r=1}\sin\frac{r\pi}{3}=\sin\frac{\pi}{3}  +\sin\frac{2\pi}{3}+\ldots+\sin\frac{8\pi}{3}=\sin  \frac{7\pi}{3}+\sin\frac{8\pi}{3}=2 \sin\frac{\pi}{3}=\sqrt{3}

    Because:
    <br />
\sin\frac{r\pi}{3}= - \sin\frac{(r+3)\pi}{3}

    -O
    Last edited by oswaldo; February 8th 2009 at 07:42 PM.
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