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Math Help - limits

  1. #1
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    limits

    how do you show that summation from x=0 to x= infinity for

    a^x / x! = exp(a)?

    thanks!
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  2. #2
    MHF Contributor
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    Quote Originally Posted by alexandrabel90 View Post
    how do you show that summation from x=0 to x= infinity for

    a^x / x! = exp(a)?

    thanks!
    Hi alexandrabel90,

    We start by assuming we can write e^a as a series of powers of a

    e^a=ba^0+ca+da^2+fa^3+ga^4+ha^5+........

    Since \frac{d}{da}e^a=e^a

    f(0)=1,\ f'(0)=1,\ f''(0)=1,\ f'''(0)=1,.....

    we successively differentiate to find all the coefficients, while setting a=0

    1=b, as\ a^0=1 and all higher powers are zero

    f'(a)=e^a=c+2da+3fa^2+4ga^3+5ha^4

    f'(0)=1=c

    f''(a)=e^a=2d+3(2)fa+4(3)ga^2+5(4)ha^3

    f''(0)=1=2d\ \Rightarrow\ d=\frac{1}{2!}

    Continuing on like this, all the co-efficients are found
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