Series Cheatsheet
MacLaurin Serie
If f a function infinitely differentiable,
+∞ (n)
X
f (0) n
x
n!
n=0
Definitions
Basic Series
Taylor’s Formula with Remainder
∃x∗ between c and x such that
Infinite Sequence: hsn i
Limit/Convergence of a Sequence: limn→∞ sn = L
P
Infinite Serie: (Partial sums) Sn = sn = s1 + s2 + · · · + sn + · · ·
Geometric Serie:
n
X
f (x) =
k=0
ark−1 = Sn = a + ar + ar2 + · · · + arn−1 =
k=1
a(1 − rn )
1−r
Rn (x) =
k!
(x − c)k + Rn (x)
f (n+1) (x∗ )
(x − c)n+1
(n + 1)!
Applications
Positive Series
Application: Showing Function/Taylor-Series Equivalence
Positive Serie: If all the terms sn are positive.
Integral Test: If f (n) = sn , continuous, positive, decreasing:
Comparison Test:
n
X
f (k) (c)
P
an and
Limit Comparison Test:
P
P
bn where ak < bk
an and
P
P
sn converges ⇐⇒
an
bn
1
exists,
P
lim Rn (x) = 0
f (x)dx converges.
P
P
1. If P bn converges, so doesP an
2. If an diverges, so does bn
(∀k ≥ m)
bn such that limn→∞
R∞
an converges ⇐⇒
P
bn converges.
n→+∞
Application: Approximating Functions or Integrals
Rn (x0 ) < K
Binomial Serie
(1 + x)r = 1 +
Convergence
Alternating Serie:
X
(−1)n+1 an = a1 − a2 + a3 − a4 + a5 − · · ·
Absolute Convergence: If
P
|sn | is convergent.
P
Conditional Convergence: If sn is convergent but not absolutely convergent.
Ratio Test: If limn→∞ | sn+1
sn | =
Root Test: If limn→∞
p
n
|sn | =
ln(1 + x) =
Power Serie About Zero:
+∞
X
(−1)n−1
xn
1
1
1
= x − x2 + x3 − x4 +
n
2
3
4
∞
X
(−1)n x2n+1
x3
x5
x7
=x−
+
−
+ ···
sin x =
(2n
+
1)!
3!
5!
7!
n=0
fn (x) − f (x) < an (x − c)n = a0 + a1 (x − c) + a2 (x − c)2 + · · ·
n=0
∞
X
n=0
• < 1: absolutely convergent
• 1: (no conclusion)
• > 1 or +∞: diverges
cos x =
∞
X
(−1)n x2n
x2
x4
x6
=1−
+
−
+ ···
(2n)!
2!
4!
6!
n=0
sinh x =
+∞
X
∞
X
x2
x3
xn
=1+x+
+
+ ···
n!
2!
3!
n=0
∞
X
1
=
xn = 1 + x + x2 + x3 + · · · +
1 − x n=0
Power Series
Power Serie:
Common Series
ex =
• < 1: absolutely convergent
• 1: (no conclusion)
• > 1 or +∞: diverges
Uniform Convergence: If ∀ > 0, ∃m such that for each x and every n ≥ m,
+∞
X
r(r − 1)(r − 2) · · · (r − n + 1) n
x
n!
n=1
∞
X
x2n+1
x3
x5
x7
=x+
+
+
+ ···
(2n
+
1)!
3!
5!
7!
n=0
cosh x =
∞
X
x2n
x2
x4
x6
=1+
+
+
+ ···
(2n)!
2!
4!
6!
n=0
an xn = a0 + a1 x + a2 x2 + · · ·
n=0
Taylor Serie
If f a function infinitely differentiable,
+∞ (n)
X
f (c)
(x − c)n
n!
n=0
Author: Martin Blais, 2009. This work is licensed under the Creative Commons “Attribution - Non-Commercial - Share-Alike” license.