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Thread: Series problem with limiting sum

  1. #1
    Member jacs's Avatar
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    Series problem with limiting sum

    A limit problem i have tried for hours now and just cannot wrap my head around
    Solution is apparently (1 + x)/(1 - x)^2

    any help greatly appreciated,
    thanks
    jacs
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  2. #2
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    We are adding all of these together. So let us add them in a convenient way.

    First we add the left-most coloums together:
    $\displaystyle 1+x+x^2+x^3+... = \frac{1}{1-x}$

    Second we add the next left-most coloums together:
    $\displaystyle 2x+2x^2+2x^3+... = 2x(1+x+x^2+...) = \frac{2x}{1-x}$

    Third we add the next left-most colums together:
    $\displaystyle 2x^2+2x^3+2x^4+... = 2x^2(1+x+x^2+...) = \frac{2x^2}{1-x}$

    And so on....

    So in total we need to add,
    $\displaystyle \frac{1}{1-x} + \frac{2x}{1-x}+\frac{2x^2}{1-x}+\frac{2x^3}{1-x}+...$

    I am going to add and subtract $\displaystyle \frac{1}{1-x}$ to make it an easier form.

    $\displaystyle \left(\frac{2}{1-x} + \frac{2x}{1-x}+\frac{2x^2}{1-x}+\frac{2x^3}{1-x}+... \right) - \frac{1}{1-x}$
    Factor again,
    $\displaystyle \frac{2}{1-x}\left( 1+x+x^2+x^3+... \right) - \frac{1}{1-x}$

    $\displaystyle \frac{2}{(1-x)^2} - \frac{1}{1-x} =\boxed{ \frac{1+x}{(1-x)^2}}$
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  3. #3
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    Hello, jacs!


    Observe that:
    . .$\displaystyle 1 \:=\:1$
    .$\displaystyle 3x\:=\:x + 2x$
    $\displaystyle 5x^2\:=\:x^2 + 2x^2 + 2x^2$
    $\displaystyle 7x^3 \:=\:x^3 + 2x^3 + 2x^3 + 2x^3$
    $\displaystyle 9x^4 \:=\:x^4 + 2x^4 + 2x^4 + 2x^4 + 2x^4$

    By studying the arrangement, or otherwise, find in simplest algebraic form,
    . . an expression for the limit of the series: .$\displaystyle 1 + 3x + 5x^2 + 7x^3 + 9x^4 + \cdots$

    Solution: .$\displaystyle \frac{1+x}{(1-x)^2}$
    Add the stack of equations . . .

    The left side is: .$\displaystyle S\:=\:1 + 3x + 5x^2 + 7x^3 + 9x^4 + \cdots$

    The right side is: .$\displaystyle (1 + x + x^2 + \cdots) + (2x + 2x^2 + 2x^3 + \cdots) + (2x^2 + 2x^3 + 2x^4+\cdots) + \cdots $

    . . . . . . . . . . .$\displaystyle = \;(1 + x + x^2 + \cdots) + 2x(1 + x + x^2 + \cdots) + 2x^2(1 + x + x^2 + \cdots) + \cdots$

    Factor: .$\displaystyle (1 + x + x^2 + \cdots)\cdot[1 + 2x + 2x^2 + 2x^3 + \cdots]$

    . . . . $\displaystyle = \;(1 + x + x^2 + \cdots)\cdot\left[1 + 2x(1 + x + x^2 + \cdots)\right]$

    The expressions in parentheses are geometric series:
    . . first term $\displaystyle a = 1$, common ratio $\displaystyle r = x$
    Their sum is: .$\displaystyle \frac{1}{1-x}$

    Hence, we have: .$\displaystyle S \;=\;\frac{1}{1-x}\cdot\left[1 + 2x\left(\frac{1}{1-x}\right)\right]$

    . . which simplifies to: .$\displaystyle \boxed{S \;=\;\frac{1+x}{(1-x)^2}}$

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Another solution . . . (They did say "otherwise", right?)


    We are given: .$\displaystyle S \;=\;1 + 3x + 5x^2 + 7x^3 + 9x^4 + \cdots$

    Multiply by $\displaystyle x\!:\;xS \;=\;\qquad x + 3x^2 + 5x^3 + 7x^4 + \cdots$

    Subtract: .$\displaystyle S - xS \;=\;1 + 2x + 2x^2 + 2x^3 + 2x^4 + \cdots$

    . . . . . . .$\displaystyle (1 - x)S \;=\; 1 + 2x\underbrace{(1 + x + x^2 + x^3 +\cdots)}_{\text{geometric series}} $

    So we have : .$\displaystyle (1 - x)S \;=\;1 + 2x\left(\frac{1}{1-x}\right) \;=\;\frac{1+x}{1-x}$

    Therefore: .$\displaystyle S \;=\;\frac{1+x}{(1-x)^2}$

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  4. #4
    Member jacs's Avatar
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    thanks to both of you. i dont think i ever woudl have figured that one out in a milion years. Going ot have a study of all the different methods and see which one i can try to replicate without giving myself and annureism.

    thanks, you guys rock!
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