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Math 101 Chapter 2: Functions of a real variable
2.26 The reciprocal hyperbolic functions
2.28 Standard limits of functions
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2.27 Exponential series
Example
The exponential series
e
x
=
∑
n
=
0
∞
x
n
n
!
=
1
+
x
+
x
2
2
!
+
x
3
3
!
+
…
e^{x}=\sum_{n=0}^{\infty}{{x^{n}}\over{n!}}=1+x+{{x^{2}}\over{2!}}+{{x^{3}}% \over{3!}}+\dots
converges for all real
x
x
.
We can also consider
e
z
=
∑
n
=
0
∞
z
n
n
!
=
1
+
z
+
z
2
2
!
+
z
3
3
!
+
…
e^{z}=\sum_{n=0}^{\infty}{{z^{n}}\over{n!}}=1+z+{{z^{2}}\over{2!}}+{{z^{3}}% \over{3!}}+\dots
for any complex number
z
z
.