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Math Help - Sum of convergent series

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    Sum of convergent series

    How do I find the sum of this?

    I got the sum of 2/3 ^n because its a Geometric sequence.

    But the other part is tricky. I separated into two partial fractions but am stuck.Sum of convergent series-imageuploadedbytapatalk-21373813938.053766.jpg
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  2. #2
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    Re: Sum of convergent series

    Quote Originally Posted by minneola24 View Post
    How do I find the sum of this?
    I got the sum of 2/3 ^n because its a Geometric sequence.
    But the other part is tricky. I separated into two partial fractions but am stuck.Click image for larger version. 

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    That is a collapsing sum: \sum\limits_{n = 1}^\infty  {\left( {\frac{1}{{(n + 1)(n + 2)}}} \right)}  = \sum\limits_{n = 1}^\infty  {\left( {\frac{1}{{(n + 1)}} - \frac{1}{{(n + 2)}}} \right)}

    Any partial sum looks like {S_K} = \sum\limits_{n = 1}^N {\left( {\frac{1}{{(n + 1)}} - \frac{1}{{(n + 2)}}} \right)}  = \frac{1}{2} - \frac{1}{{N + 2}} thus \left(S_N\right)\to~?.
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    Re: Sum of convergent series

    How did you get to .5-1/(n+1)?

    How do i complete this partial sum?
    Thanks
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    Re: Sum of convergent series

    Quote Originally Posted by minneola24 View Post
    How did you get to .5-1/(n+1)?
    How do i complete this partial sum?
    Do you know what it actually means for a infinite series to converge?
    What do partial sums have to do with?

    Do you understand how {S_N} = \sum\limits_{n = 1}^N {\left( {\frac{1}{{(n + 1)}} - \frac{1}{{(n + 2)}}} \right)}  = \frac{1}{2} - \frac{1}{{N + 2}} actually works?

    If not, there is no point in your trying the question.
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