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