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Math Help - Series and Sequences..deriving the formula

  1. #1
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    Series and Sequences..deriving the formula

    _______________________n
    Consider the summation SUM ( (1/i) - (1/(i+1)) )
    ____________________ i=1

    Derive a formula for this sum in terms of n.
    n
    Hint: SUM ( A(i) - A(i+1) )
    i=1
    = (A(1)-A(2)) + (A(2)-A(3)) + ... + (A(n)-A(n+1)) (who cancels??)
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  2. #2
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    Quote Originally Posted by kashifzaidi View Post
    _______________________n
    Consider the summation SUM ( (1/i) - (1/(i+1)) )
    ____________________ i=1

    Derive a formula for this sum in terms of n.
    n
    Hint: SUM ( A(i) - A(i+1) )
    i=1
    = (A(1)-A(2)) + (A(2)-A(3)) + ... + (A(n)-A(n+1)) (who cancels??)
    Write out the first few terms:

    \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right)  + .....

    Now simplify. What do you notice? What conclusion do you draw?
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  3. #3
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    Series and Sequences..deriving the formula-2-.jpg
    I done the following according to my notes. My book says find the common thing and divide it by 2 on both sides. I cannot find any common. I dont know how do it futher..
    Plz help me
    This is a page from my book.
    Series and Sequences..deriving the formula-properties.jpg
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  4. #4
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    The answer is simply \sum\limits_{i = 1}^n {\left( {\frac{1}<br />
{i} - \frac{1}<br />
{{i + 1}}} \right)}  = 1 - \frac{1}<br />
{{n + 1}}

    These are known as collapsing sums.
    Here only the first and last terms remain after subtraction.
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