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Section 7.2 Power Series Coefficients

To check your understanding of Section 7.1, you should complete the following activity.

Activity 7.2.1.

  1. Find the first five nonzero coefficients for \(\sin(\theta)\) expanded around the origin.

  2. Write out a series approximation, correct to fourth order, for \(\sin(\theta)\) expanded around the origin.

    \begin{equation*} \sin(\theta) = \end{equation*}

  3. Find the first four nonzero coefficients for \(\sin(\theta)\) expanded around \(\theta_0={\pi\over 6}\text{.}\)

  4. Write out a series approximation, correct to fourth order, for \(\sin(\theta)\) expanded around \(\theta_0={\pi\over 6}\text{.}\)

    \begin{equation*} \sin(\theta) = \end{equation*}
Hint

Some important observations are given below:

  1. Pay attention to the name of the independent variable. The equation for the coefficients is given in terms of the variable \(z\text{.}\) What is the independent variable in \(\sin\theta\text{?}\)

  2. Typically, there are an infinite number of terms in a power series expansion. Unless you can find a general expression for the coefficients, it will take you an infinite amount of time to find all of them. In practice, one usually stops at some stage. Some language for this: “find the first four non-zero terms” means find the coefficients for the four lowest powers of the independent variable, continuing until you have four that are not zero; “find the expansion correct to fourth order” means find the coefficients for all of the low powers of the independent variable, up to and including the fourth power.

  3. If you are asked to find the power series expansion around \(z=a\text{,}\) then you must plug the number \(a\) into all of the derivatives.

  4. Your final answer for a power series expansion should be of the form

    \begin{equation} f(z)=c_0 + c_1(z-a) + c_2(z-a)^2 + c_3(z-a)^3 + \dots\tag{7.2.1} \end{equation}

    or

    \begin{equation} f(z)\approx c_0 + c_1(z-a) + c_2(z-a)^2 + c_3(z-a)^3\tag{7.2.2} \end{equation}

    where you have plugged in numbers for all of the \(c_n\) and for \(a\text{.}\) Use the first form, with an equals sign, if you include \(\dots\) to remind yourself that there are (an infinite number of) terms that you have not written down. Use the second from, with an approximately equals sign, if you have truncated the series, leaving out (an infinite number of) terms.

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