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Math Help - Convergence of infinite series

  1. #1
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    Convergence of infinite series

    Investigate the convergence of the following infinite series and find their
    sums, where appropriate:

    i)


    ii)


    iii)

    Hint: Find explicit expressions for the partial sums SN.



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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by matty888 View Post
    Investigate the convergence of the following infinite series and find their

    sums, where appropriate:

    i)

    ii)


    iii)

    Hint: Find explicit expressions for the partial sums SN.



    I will give you a hint...in part i there is a cancellation that can be made and in part ii and iii they are telescoping series
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  3. #3
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    Quote Originally Posted by matty888 View Post
    Investigate the convergence of the following infinite series and find their

    sums, where appropriate:

    i)

    ii)


    iii)

    Hint: Find explicit expressions for the partial sums SN.
    Expanding a small bit on post #2:

    i) \frac{2^n + 3(5^n)}{5^{n+1}} = \left(\frac{1}{5}\right) \, \left( \frac{2^n}{5^n} \right) + \frac{3}{5}. It's the second term that will cause the series to diverge (why?)

    ii) Using partial fractions: \frac{4}{(4n-3)(4n+1)} = \frac{1}{4x-3} - \frac{1}{4x+1}.

    For both ii) and iii), write out the first few terms of each series and note the cancelling of terms that you can do .....
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