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Math Help - 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 secx

    2. Assuming the series for e^x and tanx, determine the series for e^xtanx up to and including the term in 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 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 x=0 and equal to \sec(0)=1

    The first derivative:

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

    which is zero at x=0

    Second derivative:

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

    which is equal to 1 at x=0

    Third derivative:

    \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 x=0

    Fourth derivative:

    \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 5 when x=0.

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

    \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):

    \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:

    \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 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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