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Math Help - sum of an infinite series

  1. #1
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    sum of an infinite series

    Find the exact sum of
    \sum_{i = 0}^{\inf} \frac{4+6^n}{8^n}


    because its an infinite series i know i have to use the formula
    \frac{a}{1-x}
    but im not sure about the general term...
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  2. #2
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    Quote Originally Posted by acosta0809 View Post
    Find the exact sum of
    \sum_{i = 0}^{\inf} \frac{4+6^n}{8^n}


    because its an infinite series i know i have to use the formula
    \frac{a}{1-x}
    but im not sure about the general term...

    \sum\limits_{n = 0}^\infty  {\frac{{4 + {6^n}}}<br />
{{{8^n}}}}  = 4\sum\limits_{n = 0}^\infty  {\frac{1}<br />
{{{8^n}}}}  + \sum\limits_{n = 0}^\infty  {\frac{{{6^n}}}<br />
{{{8^n}}}}  = 4\sum\limits_{n = 0}^\infty  {\frac{1}<br />
{{{8^n}}}}  + \sum\limits_{n = 0}^\infty  {{{\left( {\frac{3}<br />
{4}} \right)}^n}} .

    4\sum\limits_{n = 0}^\infty  {\frac{1}{{{8^n}}}}  = 4 \cdot \frac{1}{{1 - {1 \mathord{\left/{\vphantom {1 8}} \right.\kern-\nulldelimiterspace} 8}}} = 4 \cdot \frac{8}{7} = \frac{{32}}{7}.

    \sum\limits_{n = 0}^\infty  {{{\left( {\frac{3}{4}} \right)}^n}}  = \frac{1}{{1 - {3 \mathord{\left/{\vphantom {3 4}} \right.<br />
 \kern-\nulldelimiterspace} 4}}} = 4.

    \sum\limits_{n = 0}^\infty  {\frac{{4 + {6^n}}}{{{8^n}}}}  = \frac{{32}}{7} + 4 = \frac{{60}}{7}.
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