Taylor Series: e^-x near 1:

We need to expand with a Taylor series: , where is the n^th derivative of f. Since this approximates the exponential function best when x is close to c,

we take c = 1.0000.

For convenience we create the following table with c set to 1:

Taking the successive derivatives, we can fill in the middle column:

Now we evaluate the derivatives at 1:

Substitute the values from the table into the first 3 terms of the Taylor series:

which simplifies to

For , this becomes

Evaluating this, we get .