Power Series Expansion of the Error Function
Several times in this course we’ve seen the error function:
� x
2
2
erf(x) = √
e−t dt.
π 0
(The factor of √2π guarantees that limx→∞ erf(x) = 1.) This function is very
important in probability theory, but we don’t have a conventional algebraic
description of it.
Because we can integrate power series and know that:
2
e−t = 1 − t2 +
t4
t6
− + ···
2! 3!
(R = ∞),
we can now find a power series expansion for the error function.
� x
2
2
erf(x) = √
e−t dt
π 0
� x
2
t4
t6
= √
(1 − t2 + − + · · ·)dt
2! 3!
π 0
�
�x
3
5
2
t
t
t7
√
−
=
t− +
+ · · ·
5 · 2! 7 · 3!
3
π
0
�
�
2
x3
x5
x7
−
= √
x−
+
+ ···
5 · 2! 7 · 3!
3
π
To get this to look exactly like a power series we would distribute the factor of
√2 across the sum, multiplying it by each term of the series. However, that’s
π
not strictly necessary.
This turns out to be a very good way to compute the value of the error
function; your calculator probably uses this method.
1
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18.01SC Single Variable Calculus��
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