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  1. #1
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    summations

    Well, I had a summer homework packet that I put off, and now I really need to finish it fast... Anyway, I know how summation notation works, but in my packet there's a problem that says "Find Each summation" and where there's usually a number over the sigma letter, there's an infinity symbol. I never learned this last year, so I don't know what to do. Help?
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by iplayaltosax View Post
    Well, I had a summer homework packet that I put off, and now I really need to finish it fast... Anyway, I know how summation notation works, but in my packet there's a problem that says "Find Each summation" and where there's usually a number over the sigma letter, there's an infinity symbol. I never learned this last year, so I don't know what to do. Help?
    It represents an infinite series. What is the exact problem? Post it here so we have a better idea of how to help you.

    --Chris
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    Quote Originally Posted by iplayaltosax View Post
    Well, I had a summer homework packet that I put off, and now I really need to finish it fast... Anyway, I know how summation notation works, but in my packet there's a problem that says "Find Each summation" and where there's usually a number over the sigma letter, there's an infinity symbol. I never learned this last year, so I don't know what to do. Help?
    there are various ways to tackle problems like these based on the summation you have. it would be good for you to state the exact problem. otherwise we might go into explaining things that are not necessary to explain, re-inventing the wheel, etc
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    Well, below the sigma letter it says i=1, then off to the right it says 3(1/2)^(i-1). That better?
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by iplayaltosax View Post
    Well, below the sigma letter it says i=1, then off to the right it says 3(1/2)^(i-1). That better?
    \sum_{i=1}^{\infty}3\left(\frac{1}{2}\right)^{i-1}

    This is a geometric series, where a=3 and r=\frac{1}{2}

    Since \left|\frac{1}{2}\right|<1, the series will converge to a finite sum, and that sum is \frac{a}{1-r}

    Try to take it from here.

    --Chris
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    Quote Originally Posted by Chris L T521 View Post
    \sum_{i=1}^{\infty}3\left(\frac{1}{2}\right)^{i-1}

    This is a geometric series, where a=3 and r=\frac{1}{2}

    Since \left|\frac{1}{2}\right|<1, the series will converge to a finite sum, and that sum is \frac{a}{1-r}

    Try to take it from here.

    --Chris
    Thanks, that helped on that one. I get it, but the next problem says

    \sum_{i=1}^{n}(i^2 -3i+2)<br />
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by iplayaltosax View Post
    Thanks, that helped on that one. I get it, but the next problem says

    \sum_{i=1}^{n}(i^2 -3i+2)<br />
    recall that the summation of a sum is just the sum of the summations. and also that we can factor constants out and put them in front of the summation symbol. thus,

    \sum_{i = 1}^n (i^2 - 3i + 2) = \sum_{i = 1}^ni^2 -3 \sum_{i = 1}^ni + \sum_{i = 1}^n2

    and recall the rules:

    \sum_{i = 1}^ni^2 = \frac {n(n + 1)(2n + 1)}6, \sum_{i = 1}^ni = \frac {n(n + 1)}2, and \sum_{i = 1}^nc = cn for c a constant

    i think you can take it from here
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  8. #8
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    Hello !
    Quote Originally Posted by iplayaltosax View Post
    Thanks, that helped on that one. I get it, but the next problem says

    \sum_{i=1}^{n}(i^2 -3i+2)<br />
    Hmmm I have 2 possibilities for this... The second one being unsure, it's the first time

    1st :
    Note that \sum_{i=1}^n (i^2-3i+2)=\sum_{i=1}^n i^2-3 \sum_{i=1}^n i+2 \sum_{i=1}^n 1

    Formulae :
    \sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}

    \sum_{k=1}^n k=\frac{n(n+1)}{2}


    -------------------------------------------
    2nd : you gotta be used to summations, so don't read if you don't want to... The first part is largely enough
    i^2-3i+2=(i-2)(i-1)

    S=\sum_{i=1}^n (i^2-3i+2)=\sum_{i=1}^n (i-2)(i-1)

    We'll try to write this sum in the form \sum_{i=\dots}^{\dots} (i-1)(i)
    Change i into t+1.
    i-2 --> t-1
    i-1 --> t
    i=1 --> t=0
    i=n --> t=(n-1)
    Then S=\sum_{i=1}^n (i-2)(i-1)=\sum_{t=0}^{n-1} (t-1) \cdot t


    Thus \begin{aligned} 2S &=\sum_{i=1}^n (i-2)(i-1)+\sum_{t=0}^{n-1} (t-1) \cdot t \\<br />
&=\sum_{i=1}^n (i-2)(i-1)+\sum_{i=0}^{n-1} (i-1) \cdot i \\<br />
&=\left(\sum_{i=1}^{{\color{red}n-1}} (i-2)(i-1)\right)+{\color{red}(n-2)(n-1)}+\left(\sum_{{\color{red}i=1}}^{n-1} (i-1) \cdot i\right)+{\color{red}0} \end{aligned}


    2S=(n-2)(n-1)+\sum_{i=1}^{n-1} \bigg[(i-2)(i-1)+(i-1)i\bigg]

    2S=(n-2)(n-1)+\sum_{i=1}^{n-1} (i-1)\underbrace{(i-2+i)}_{2(i-1)}

    2S=(n-2)(n-1)+2 \sum_{i=1}^{n-1} (i-1)^2

    Changing i again :

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

    2S=(n-2)(n-1)+2 \left(\frac{(n-2)(n-1)(2n-3)}{6}\right) By applying the formula above.

    2S=(n-2)(n-1) \left(1+\frac{2n-3}{3}\right)

    And so on... it's much longer than the normal method



    Edit : no wonder I'm too late -___-' what an idea to make such a useless stuff lol !
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  9. #9
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    Quote Originally Posted by Jhevon View Post
    recall that the summation of a sum is just the sum of the summations. and also that we can factor constants out and put them in front of the summation symbol. thus,

    \sum_{i = 1}^n (i^2 - 3i + 2) = \sum_{i = 1}^ni^2 -3 \sum_{i = 1}^ni + \sum_{i = 1}^n2

    and recall the rules:

    \sum_{i = 1}^ni^2 = \frac {n(n + 1)(2n + 1)}6, \sum_{i = 1}^ni = \frac {n(n + 1)}2, and \sum_{i = 1}^nc = cn for c a constant

    i think you can take it from here
    Wow. Thanks a lot. I'm positive that I never learned this in the school year, so I was wondering where I could find these rules and maybe some explanations... Would you know where?
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  10. #10
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by iplayaltosax View Post
    Wow. Thanks a lot. I'm positive that I never learned this in the school year, so I was wondering where I could find these rules and maybe some explanations... Would you know where?
    well, if you want to find out about summation rules, googling "summation rules" is always a good idea. here is a page i found by doing that: http://www.math.ucdavis.edu/~kouba/C...Summation.html

    otherwise you can check well known sources like wikipedia.org, and maybe it is on purplemath
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