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Thread: Multiplication of Taylor series

  1. #1
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    Multiplication of Taylor series

    The question is:

    Use multiplication of Taylor series to find the quartic Taylor polynomial about 0 for the function:

    $\displaystyle f(x)=\frac{sinx}{\sqrt{1+x}}$ evaluating the coefficients.

    For sin x the standard Taylor series about 0 is:
    $\displaystyle x-\frac{1}{3!}x^3+...$

    For $\displaystyle \sqrt{1+x}$ this can be rearranged to fit the standard Talor series
    $\displaystyle (1+x)^\alpha$, which is

    $\displaystyle 1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3$ ...

    Multiplying these two together the answer I get is

    $\displaystyle \frac{1}{2}x+\frac{3}{8}x^2+\frac{5}{96}x^3+\frac{ 35}{128}x^4$

    Does this look along the right lines?
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  2. #2
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    No, that's incorrect, I'm afraid. You need to get the series for $\displaystyle \frac{1}{\sqrt{1+x}},$ not $\displaystyle \sqrt{1+x}.$ Multiply those two series together to get the result.
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  3. #3
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    Quote Originally Posted by Ackbeet View Post
    No, that's incorrect, I'm afraid. You need to get the series for $\displaystyle \frac{1}{\sqrt{1+x}},$ not $\displaystyle \sqrt{1+x}.$ Multiply those two series together to get the result.
    We have a list of 6 standard Taylor series about 0 that we have been told to use. There isn't one for $\displaystyle \frac{1}{\sqrt{1+x}}$, which is why I used the one for $\displaystyle (1+x)^\alpha$
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  4. #4
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    Well, then use $\displaystyle \alpha=-\frac{1}{2}.$ Is that what you used?
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  5. #5
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    Quote Originally Posted by Ackbeet View Post
    Well, then use $\displaystyle \alpha=-\frac{1}{2}.$ Is that what you used?
    That's what I used, but haven't got the correct answer. In the next part of the question we have to use a software programme that we're given to check we have the correct answer, so I know what it should be, but can't seem to get it right
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  6. #6
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    Hmm. We have

    $\displaystyle \displaystyle{\sin(x)=x-\frac{x^{3}}{3!}+\dots,$ as you had before. We don't need more terms because we're only looking for the first four terms. According to your formula there, we have

    $\displaystyle \displaystyle{(1+x)^{\alpha}=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3},$

    so

    $\displaystyle \displaystyle{(1+x)^{-1/2}=1+(-1/2) x+\frac{(-1/2)(-1/2-1)}{2!}x^2+\frac{(-1/2)(-1/2-1)(-1/2-2)}{3!}x^3+\dots}$
    $\displaystyle \displaystyle{=1-x/2+\frac{3x^{2}}{8}-\frac{5 x^{3}}{16}+\dots}$

    We don't need more terms than these, because the lowest power of x in the sin series is to the first power; hence, the cubic in this series will go to a fourth power.

    Both of these are correct. Moving on, then:

    $\displaystyle \displaystyle{\frac{\sin(x)}{\sqrt{1+x}}=\left(x-\frac{x^{3}}{3!}+\dots\right)\left(1-x/2+\frac{3x^{2}}{8}-\frac{5 x^{3}}{16}+\dots\right)}$

    $\displaystyle \displaystyle{=x-\frac{x^{2}}{2}+\frac{3x^{3}}{8}-\frac{5x^{4}}{16}-\frac{x^{3}}{6}+\frac{x^{4}}{12}+\dots}$

    $\displaystyle \displaystyle{=x-\frac{x^{2}}{2}+\frac{5x^{3}}{24}-\frac{11x^{4}}{48}+\dots}$

    Can you see where your mistake is now?
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  7. #7
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    Quote Originally Posted by Ackbeet View Post
    Hmm. We have

    $\displaystyle \displaystyle{\sin(x)=x-\frac{x^{3}}{3!}+\dots,$ as you had before. We don't need more terms because we're only looking for the first four terms. According to your formula there, we have

    $\displaystyle \displaystyle{(1+x)^{\alpha}=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3},$

    so

    $\displaystyle \displaystyle{(1+x)^{-1/2}=1+(-1/2) x+\frac{(-1/2)(-1/2-1)}{2!}x^2+\frac{(-1/2)(-1/2-1)(-1/2-2)}{3!}x^3+\dots}$
    $\displaystyle \displaystyle{=1-x/2+\frac{3x^{2}}{8}-\frac{5 x^{3}}{16}+\dots}$

    We don't need more terms than these, because the lowest power of x in the sin series is to the first power; hence, the cubic in this series will go to a fourth power.

    Both of these are correct. Moving on, then:

    $\displaystyle \displaystyle{\frac{\sin(x)}{\sqrt{1+x}}=\left(x-\frac{x^{3}}{3!}+\dots\right)\left(1-x/2+\frac{3x^{2}}{8}-\frac{5 x^{3}}{16}+\dots\right)}$

    $\displaystyle \displaystyle{=x-\frac{x^{2}}{2}+\frac{3x^{3}}{8}-\frac{5x^{4}}{16}-\frac{x^{3}}{6}+\frac{x^{4}}{12}+\dots}$

    $\displaystyle \displaystyle{=x-\frac{x^{2}}{2}+\frac{5x^{3}}{24}-\frac{11x^{4}}{48}+\dots}$

    Can you see where your mistake is now?
    Thank you so much for all your help. just one final question (I promise!). I worked out that I needed to take away $\displaystyle \frac{1}{6}x^3$ from $\displaystyle \frac{3}{8}x^3$ to get the correct coefficient and add $\displaystyle \frac{1}{12}x^4$to $\displaystyle \frac{-5}{16}x^4$, but I'm not sure where the $\displaystyle \frac{1}{12}x^4$ comes from? Thanks again
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  8. #8
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    The $\displaystyle \frac{x^{4}}{12}$ comes from multiplying the $\displaystyle -\frac{x^{3}}{3!}=-\frac{x^{3}}{6}$ in the sin expansion with the $\displaystyle -\frac{x}{2}$ in the square root expansion.
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