174 A. USEFUL FORMULAS
RATIO TEST
Suppose that
1
X
nD0
x
n
is a series and r D lim
n!1
ˇ
ˇ
ˇ
ˇ
x
nC1
x
n
ˇ
ˇ
ˇ
ˇ
. en
• if r < 1, then the series converges
• if r > 1, then the series diverges
• if r D 1, then the test is inconclusive
ROOT TEST
Suppose that
1
X
nD0
x
n
is a series and s D lim
n!1
n
p
jx
n
j: en
• if s < 1, then the series converges
• if s > 1, then the series diverges
• if s D 1, then the test is inconclusive
A.10 TAYLOR SERIES
If f .x/ is a function that can be differentiated any number of times,
f .x/ D
1
X
nD0
f
.n/
.c/.x c/
n
nŠ
SPECIAL TAYLOR SERIES
• e
x
D
1
X
nD0
x
n
nŠ
• sin.x/ D
1
X
nD0
.1/
n
x
2nC1
.2n C1/Š
• cos.x/ D
1
X
nD0
.1/
n
x
2n
.2n/Š
TAYLOR’S INEQUALITY
Suppose we are looking at a Taylor polynomial for the function f .x/ in the interval jx aj d
and that jf
.nC1/
.x/j M everywhere in this interval. en for values of x in the interval we
have that
jR
n
.x/j
M
.n C1/Š
jx aj
nC1
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