Math Help - limits

1. limits

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

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

thanks!

2. Originally Posted by alexandrabel90
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