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Math Help - limit

  1. #1
    Newbie
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    limit

    Hi,

    need some help with this problem

    a. Show that, whatever is the whole number, k is superior or equal to 2;

    1/k^2 is less than or equal to 1/ k-1 - 1/ K

    Have done this and think I have the answer but part 2 and 3 are complicated

    b. Use this result to show the inequality

    n 1
    € ----- is less than or equal to
    k=2 K^2

    n
    c. Show that (Un) which is defined by Un=€ .1...
    k=2 K^2 is convergent

    Thanks in advance for any hints to proceed...
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  2. #2
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    \begin{gathered}<br />
  \left( {\forall n \geqslant 2} \right)\left[ {\frac{1}<br />
{{n^2 }} \leqslant \frac{1}<br />
{{n - 1}} - \frac{1}<br />
{n}} \right] \hfill \\<br />
  \sum\limits_{n = 2}^K {\frac{1}<br />
{{n^2 }}}  \leqslant n = \sum\limits_{n = 2}^K {\left[ {\frac{1}<br />
{{n - 1}} - \frac{1}<br />
{n}} \right]}  = 1 - \frac{1}<br />
{K} \hfill \\ <br />
\end{gathered}
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