The error function defined by

gives the probability that any one of a series of trials will lie within x units of the mean, assuming that the trials have a normal distribution with mean 0 and standard deviationˆš2/2. This integral cannot be evaluated in terms of elementary functions, so an approximating technique must be used.
a. Integrate the Maclaurin series for eˆ’x2 to show that

b. The error function can also be expressed in the form

Verify that the two series agree for k = 1, 2, 3, and 4. [Use the Maclaurin series for (eˆ’x)2 .]
c. Use the series in part (a) to approximate erf(1) to within 10ˆ’7.
d. Use the same number of terms as in part (c) to approximate erf(1) with the series in part (b).
e. Explain why difficulties occur using the series in part (b) to approximate erf(x).

VT erf(x) = - (2k + 1)k! -o 1-3-5.. (2k 1)