Exploration 3
A Series with a Curious Property
1.
f
9
(
x
)
5
1
1
x
1 }
x
2
2
!
} 1 }
x
3
3
!
} 1
…
1 }
x
n
n
!
} 1
…
.
2.
f
(0)
5
1
1
0
1
0
1
…
5
1.
3.
Since this function is its own derivative and takes on the
value 1 at
x
5
0, we suspect that it must be
e
x
.
4.
If
y
5
f
(
x
), then
}
d
d
y
x
} 5
y
and
y
5
1 when
x
5
0.
5.
The differential equation is separable.
}
d
y
y
} 5
dx
E
}
d
y
y
} 5
E
dx
ln
)
y
)
5
x
1
C
y
5
Ke
k
1
5
Ke
0
⇒
K
5
1
[
y
5
e
x
.
6.
The first three partial sums are shown in the graph below. It
is risky to draw any conclusions about the interval of
convergence from just three partial sums, but so far the
convergence to the graph of
y
5
e
x
only looks good on
(
2
1, 1). Your answer might differ.
[
2
5, 5] by [
2
3, 3]
7.
The next three partial sums show that the convergence
extends outside the interval (
2
1, 1) in both directions, so
(
2
1, 1) was apparently an underestimate. Your answer in
#6 might have been better, but unless you guessed “all real
numbers,” you still underestimated! (See Example 3 in
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 Spring '08
 GERMAN
 Calculus, Derivative, lim, partial sums

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