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Derivative of cosh x

  1. Jan 10, 2013 #1
    maths.png

    Hi... I read this from my lecture note, but i couldn't understand:

    i) the red part, shouldn't it be k =0 instead of k=1?

    ii) the third line to the fourth line... i.e.

    maths 2.png

    I have no idea how to change from the third line to the fourth line..

    Can anyone enlighten me? Thanks a lot.. :smile:
     
  2. jcsd
  3. Jan 10, 2013 #2

    HallsofIvy

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    It doesn't matter. Since the terms being summed have a factor of "k", the k= 0 term has value 0.

    In general, for power series, [itex]f(x)= \sum_{k=0}^\infty a_kx^k[/itex], the derivative can be written as either [itex]f'(x)= \sum_{k=0}^\infty ka_kx^{k- 1}[/itex] or as [itex]f'(x)= \sum_{k=1}^\infty ka_kx^{k- 1}[/itex] because the k= 0 term is 0.

    By making the change of index, n= k- 1, so that k= n+1, the second can be written [itex]f'(x)= \sum_{n=0} (n+1)a_{n+1}x^n[/itex].
     
  4. Jan 16, 2013 #3
    The first sum can be rewritten as
    [itex]\displaystyle \sum_{k=1}^\infty (2k)\frac{x^{2k-1}}{(2k)!}=\sum_{k=1}^\infty \frac{x^{2k-1}}{(2k-1)!}[/itex]​
    since [itex](2k)!=(2k)(2k-1)![/itex]. To change the base index, replace [itex]k[/itex] by [itex]k+1[/itex]. You then get the sum
    [itex]\displaystyle \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}.[/itex]​
     
  5. Jan 17, 2013 #4

    mathman

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    This seems to be a very roundabout way of getting this result.

    cosh(x) = (ex + e-x)/2
    The derivative = (ex - e-x)/2 =sinh(x).
     
  6. Jan 17, 2013 #5

    HallsofIvy

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    IF cosh(x) has been defined as [itex](e^x+ e^{-x})/2[/itex], yes, that is more direct. However, if cosh(x) has been defined as [itex]\sum_{i=0}^\infty x^{2i}/(2i)![/itex], as is quite possible and apparently as done in the first post, the argument given is more direct.
     
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