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Math Help - Evaluating Infinite Sums

  1. #1
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    Evaluating Infinite Sums

    Evaluate the following infinite sums. (In most cases they are f(a) where a is some obvious number and f(x) given by some power series. To evaluate the various power series, manipulate them until some well-known power series emerge.)

    [sum from n=0 to inf.] n/(2^n)

    Appreciate any help
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  2. #2
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    <br />
f\left( x \right) = \sum\limits_{n = 0}^\infty  {x^n }  = \frac{1}<br />
{{1 - x}}{\text{ }}\forall \left| x \right| < 1{\text{ }} \Rightarrow f'\left( x \right) = \sum\limits_{n = 0}^\infty  {nx^{n - 1} }  = \frac{1}<br />
{{\left( {1 - x} \right)^2 }}<br />

    Now you put x=\dfrac{1}{2} and you make some repare
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  3. #3
    Super Member General's Avatar
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    Quote Originally Posted by Nacho View Post
    <br />
f\left( x \right) = \sum\limits_{n = 0}^\infty {x^n } = \frac{1}<br />
{{1 - x}}{\text{ }}\forall \left| x \right| < 1{\text{ }} \Rightarrow f'\left( x \right) = \sum\limits_{{\color{red}n = 1}}^{\infty} {nx^{n - 1} } = \frac{1}<br />
{{\left( {1 - x} \right)^2 }}<br />

    Now you put x=\dfrac{1}{2} and you make some repare
    Correction.
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  4. #4
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    Hello, blorpinbloo!

    Evaluate: . \sum^{\infty}_{n=0} \frac{n}<br />
{2^n}


     \begin{array}{cccccc}<br />
\text{We have:} & S &=& \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \dfrac{5}{2^5} + \hdots \\ \\[-3mm]<br /> <br />
\text{Multiply by }\frac{1}{2}: & \dfrac{1}{2}S &=& \quad\;\; \dfrac{1}{2^2} + \dfrac{2}{2^3} + \dfrac{3}{2^4} + \dfrac{4}{2^5} + \hdots <br />
\end{array}

    . . Subtract: . . \frac{1}{2}S \;=\;\underbrace{\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \frac{1}{2^5} + \hdots}_{\text{geometric series}} .[1]


    The geometric series has the sum: . \frac{\frac{1}{2}}{1-\frac{1}{2}} \;=\;1


    Hence, [1] becomes: . \frac{1}{2}S \:=\:1


    Therefore: . S \;=\;2

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  5. #5
    Member Nacho's Avatar
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    Quote Originally Posted by General View Post
    Correction.
    Thanks, but I think that is not important, because the first terme of the sum is zero, then is same if the sum beginning from zero or one
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