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Let us say we want to have a function such that it goes through the point (0, 1) and that it is its one derivative i.e. that *y*´=*y*.

Next we assume it can be written as a power series.

The derivative of this is

Since *y*´=*y* for varying values of x, the two equations must be equal term by term, and thus the coefficients too. We get

Now *a*_{1} must be one since we want the function to go through the point (0, 1), and that means that *a*_{1}=1 and that *a*_{1}=1=2*a*_{2, }and thus that *a*_{2}=1/2, then we get that *a*_{2}=1/2=3*a*_{3}, so *a*_{3}=1/6=1/3! and so on. We thus have *a*_{n}=1/*a*!. This gives us

I.e. the exponential function *e ^{x}*, as expected.

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