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Unformatted text preview: ctor this out of the limit as: ( x) k 1
e z ( x) k 1
( x) k 1
x
lim lim e lim
k ( k 1)!
k ( k 1)!
k ( k 1)!
We should note that the limits (on the left/right) go to 0 as eventually as when we consider the series:
k 1 ( x) (k 1)!
k 0 We know that the ratio test tells us it is convergent (limit goes to 0). By the divergent series test, the kth term must also go to 0 when it is convergent.
This means we get: e z ( x) k 1
0 lim ex 0
k ( k 1)!
e z ( x) k 1
0 lim
0
k ( k 1)! Squeeze theorem tells us that the limit must then go to 0. Example 1 Continued
When x < 0, we have x < z < 0 which gives ex ≤ ez ≤ 1, making our squeeze: ( x) k 1
e z ( x) k 1
( x) k 1
e lim lim lim
k ( k 1)!
k ( k 1)!
k ( k 1)!
x Again, both limits go to zero, this makes the interior go to 0 by the squeeze theorem. This means that no matter what x is, the Rk will go to 0, and thus the series will be the exact same thing as the funct...
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This note was uploaded on 02/07/2014 for the course MATH 1004 taught by Professor Mark during the Summer '00 term at Carleton CA.
 Summer '00
 Mark
 Calculus, Approximation, Taylor Series, Lecture Notes, MATH1004, Kyle Harvey

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