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Exercises

  1. Verify that substitution and multiplication work as described above to generate Taylor series (with base point a=0) for the following functions. That is, compare the Taylor polynomials for various orders obtained directly with those obtained by substitution, multiplication, or division.
    1. $f(x) = e^{2x}$.
    2. $f(x) = x e^{-x^2}$.
    3. $f(x) = (1-\cos(x))/x^2$.
    4. $f(x) = x^2/(1-x)$.
    5. $f(x) = x^2\sin(3x)$.

  2. Use substitution followed by integration to generate the first three terms in the Taylor series with base point $a=0$ for $\arctan(x)$. Start with the series for $1/(1-x)$.

  3. Can you find the sum of the following series?

    \begin{displaymath}\sum_{k=1}^{\infty} k x^{k} \end{displaymath}

    (Hint - differentiate the series for $1/(1-x)$, then multiply by a power of $x$.)

  4. In the background section we only considered multiplication of series by polynomials. Suppose you wanted to generate the Taylor polynomial of order ten with base point $x=0$ for the function $\displaystyle h(x)=\frac{\exp(-x)}{1-x}$. Can you do this by multiplying Taylor polynomials for $\exp(-x)$ and $\displaystyle \frac{1}{1-x}$?



William W. Farr
2006-04-10