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Math Help - finding the series representation

  1. #1
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    finding the series representation

    ok the first problem i had to construct a power series and i feel pretty good about that one, but please check my work anyways. the second problem is the one really giving me fits or maybe im jus tired. basically i was going to jus take the e to the x function and subtract 1 from it and then divide by x? well i was trying to simply it so i could take the integral of each term, can you tell me where i went wrong. thanks in advance.IMG_0001.pdf
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  2. #2
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    Quote Originally Posted by slapmaxwell1 View Post
    ok the first problem i had to construct a power series and i feel pretty good about that one, but please check my work anyways. the second problem is the one really giving me fits or maybe im jus tired. basically i was going to jus take the e to the x function and subtract 1 from it and then divide by x? well i was trying to simply it so i could take the integral of each term, can you tell me where i went wrong. thanks in advance.IMG_0001.pdf
    The second one you don't need to expand  \frac{1}{x} , just expand  e^x - 1 = ( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + .... ) - 1 =  x + \frac{x^2}{2!} + \frac{x^3}{3!} + ....

    Then the integral  \int_0^x \frac{e^x-1}{x}~dx =

     \int_0^x 1 + \frac{x}{2!} + \frac{x^2}{3!} + .... ~dx

     = x +  \frac{x^2}{2\cdot 2!} + \frac{x^3}{3\cdot 3!} + ....
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  3. #3
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    why dont i need to expand 1/x? thanks again for your response...
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  4. #4
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    Because \frac{1}{x}(x+ \frac{x^2}{2!}+ \frac{x^3}{3!}+ \cdot\cdot\cdot)= 1+ \frac{x}{2!}+ \frac{x^2}{3!}+ \cdot\cdot\cdot. Leaving \frac{1}{x} in that form makes the rest of the calculation easier.
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  5. #5
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    ok i was thinking i had to integrate before i could write it in terms of x. ok i think im going to need to practice that more. now the problem never said when to stop, so how do i know how many terms to write out? and thank you for your response...
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