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Thread: series

  1. #1
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    series

    hi everyone!

    needed some help on two questions.

    1. Find the first 3 non-zero terms in the series for $\displaystyle secx$

    2. Assuming the series for $\displaystyle e^x$ and $\displaystyle tanx$, determine the series for $\displaystyle e^xtanx$ up to and including the term in $\displaystyle x^4$


    thank you.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Enita View Post
    1. Find the first 3 non-zero terms in the series for $\displaystyle secx$
    Well we need the Taylor series about 0.

    So lets look at the sequence of derivatives and find the first three which are non-zero.

    First the zero-th drrivative is non-zeros at $\displaystyle x=0$ and equal to $\displaystyle \sec(0)=1$

    The first derivative:

    $\displaystyle \frac{d \sec(x)}{dx}=\frac{\sin(x)}{\cos^2(x)}$

    which is zero at $\displaystyle x=0$

    Second derivative:

    $\displaystyle \frac{d^2 \sec(x)}{dx^2}=\frac{\sin^2(x)+1}{\cos^3(x)}$

    which is equal to $\displaystyle 1$ at $\displaystyle x=0$

    Third derivative:

    $\displaystyle \frac{d^3 \sec(x)}{dx^3}=\frac{2 \sin(x)}{\cos^2(x)}+\frac{3 \sin^3(x)+3\sin(x)}{\cos^4(x)}$

    which is zero at $\displaystyle x=0$

    Fourth derivative:

    $\displaystyle \frac{d^4 \sec(x)}{dx^4}=\frac{11 \sin^2(x)+5}{\cos^3(x)}+\frac{12 \sin^4(x)+12\sin^2(x)}{\cos^5(x)}$

    which is $\displaystyle 5$ when $\displaystyle x=0$.

    So the series expansion of $\displaystyle \sec(x)$ to the first three non zero terms is:

    $\displaystyle \sec(x)=\sec(0)+ \left. \frac{d^2 \sec(x)}{dx^2}\right|_{x=0}x^2/2 + \left. \frac{d^4 \sec(x)}{dx^4}\right|_{x=0}x^4/24+...$

    so (if I have done this right):

    $\displaystyle \sec(x)=\sec(0)+ x^2/2+(5/24)x^4+...$

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    Third derivative:

    $\displaystyle \frac{d^3 \sec(x)}{dx^3}=\frac{2 \sin(x)}{\cos^2(x)}+\frac{3 \sin^3(x)+3\sin(x)}{\cos^4(x)}$

    which is zero at $\displaystyle x=0$
    hey thank you.

    yes the answer is right checked the back of the book.

    but what do you do for the line i quoted?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Enita View Post
    hey thank you.

    yes the answer is right checked the back of the book.

    but what do you do for the line i quoted?
    It does not contribute to the series as it is zero at x=0, but we need it
    to construct the next higher order derivative.

    RonL
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