Substitution of Power Series
2
We can find the power series of e−t by starting with the power series for ex
and making the substitution x = −t2 .
ex
e−t
2
=
=
=
x2
x3
+
+ · · · (R = ∞)
2!
3!
(−t2 )2
(−t2 )3
1 + (−t2 ) +
+
+ ···
2!
3!
4
6
t
t
1 − t2 + − + · · ·
2! 3!
1+x+
The signs of the terms alternate, the powers are all even, and the denominators
are the factorials shown. The radius of convergence is infinity.
1
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18.01SC Single Variable Calculus��
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