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Math Help - Using Series (MacLaurin/Taylor)

  1. #1
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    Using Series (MacLaurin/Taylor)

    Hey all,

    My math prof is giving us an opportunity to earn back some points on our midterms, so I cam to ask for a spot of help on some problems that I missed some stuff on, hopefully you guys can give me a hand:

    #1) Finding the exact value of this series:

    The sum from k = 0 to infinity of (2^(2k+1))/(5^(k))

    When I did this out, I think that I said it added up to something, as after the first couple terms, they began to be less than one, but I guess that I was mistaken as to the actual value...any thoughts?

    #2) Compute the MacLaurin Series for e^x to compute the following:

    Indefinite Integral of (e^(x^2))/(x) as an infinite series.

    Here, I didn't really get anywhere, and he cautioned the class that you had to actually split off a term, leaving the index for the series representation to be 1 instead of zero. I wasn't exactly sure as to how we're supposed to go about doing that...

    #3) Find the 2nd Taylor polynomial, T2 (x), for f (x) = tan x centered at a= 0. What’s the maximum error in this approximation if 0 ≤ x ≤ π 6 ? (Hint: both sec x and tan x are increasing functions on the given interval.)

    For this one, I got pretty far, only missing the last step or so, but I guess that it was off by a bit, and I was wondering if someone could walk through the steps so that I could see where I went wrong with this. (I think I had all of the derivatives and consequent values right, but not sure about the rest)


    Thanks in advance.

    -B
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  2. #2
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    Quote Originally Posted by TyrsFromAbove37 View Post
    Hey all,

    My math prof is giving us an opportunity to earn back some points on our midterms, so I cam to ask for a spot of help on some problems that I missed some stuff on, hopefully you guys can give me a hand:

    #1) Finding the exact value of this series:

    The sum from k = 0 to infinity of (2^(2k+1))/(5^(k))

    When I did this out, I think that I said it added up to something, as after the first couple terms, they began to be less than one, but I guess that I was mistaken as to the actual value...any thoughts?

    #2) Compute the MacLaurin Series for e^x to compute the following:

    Indefinite Integral of (e^(x^2))/(x) as an infinite series.

    Here, I didn't really get anywhere, and he cautioned the class that you had to actually split off a term, leaving the index for the series representation to be 1 instead of zero. I wasn't exactly sure as to how we're supposed to go about doing that...

    #3) Find the 2nd Taylor polynomial, T2 (x), for f (x) = tan x centered at a= 0. What’s the maximum error in this approximation if 0 ≤ x ≤ π 6 ? (Hint: both sec x and tan x are increasing functions on the given interval.)
    1. hint ...

    \frac{2^{2k+1}}{5^k} = \frac{2 \cdot 2^{2k}}{5^k} = \frac{2 \cdot 4^k}{5^k} = 2 \left(\frac{4}{5}\right)^k

    2. e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...

    e^{x^2} = 1 + x^2 + \frac{x^4}{2!} + \frac{x^6}{3!} + ...

    so, \frac{e^{x^2}}{x} = ?

    3. the maclaurin series for tanx has only odd degree terms (because tanx is an odd function). so, the second degree taylor polynomial is just ...

    \tan{x} \approx x

    error = |\tan{x} - x| < \frac{Mx^3}{3!}

    where M = maximum value of the third derivative of \tan{x} over the given interval.
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