Math 1321-004 (Qinghai Zhang)
1
Memorization sheet for Exam 1
Basic Concepts
3
Definition 1. A sequence is simply a countable set
{an | n ∈ N+ } or a function defined on N+ .
2013-JAN-29
Remainder estimation
Definition 13. The nth remainder of the series
is defined as
Definition 2. A sequence {an } has the limit L,
lim an = L, or an → L as n → ∞,
n→∞
Rn =
(1)
∞
X
ai −
i=1
n
X
ai =
i=1
∞
X
ai .
P∞
i=1
ai
(4)
i=n+1
if
Theorem 14 (Remainder estimate for integral test).
(2) Let f be a function that satisfies the conditions in TheIf such a limit L exists, we say that {an } converges to L. orem 7, then ∀n > M ,
P∞
Z ∞
Z ∞
n−1
Theorem 3. The geometric series
with
n=1 ar
f
(x)dx
≤
R
≤
f (x)dx.
(5)
n
a 6= 0 is convergentPif |r| < 1 and divergent otherwise.
n+1
n
∞
a
n−1
In the former case, n=1 ar
= 1−r .
P∞ 1
Theorem 15 (Alternating series estimation). A converTheorem 4. The p-series
n=1 np is convergent if
gent alternating series defined in Theorem 10 satisfies
p > 1 and divergent if p ≤ 1.
|Rn | ≤ bn+1 .
∀ > 0, ∃N ∈ N, s.t. n > N ⇒ |an − L| < .
2
Convergence tests
Theorem 5.
P∞
n=1
4
an converges ⇒ limn→∞ an = 0.
Power series
Definition 16. A power series centered at a is a series
Theorem 6 (Test for divergence).
P∞If limn→∞ an does of the form
not exist or limn→∞ an 6= 0, then n=1 an diverges.
∞
X
p(x) =
cn (x − a)n ,
(6)
Theorem 7 (Integral test). Let f : R → R be a continn=0
uous bounded function satisfying f (n) = an .
0
If ∃MP∈ N+ , s.t. ∀x ∈ [M, +∞),
R ∞ f (x) > 0 and f (x) < 0, where cn ’s are the coefficients. The interval of conver∞
then n=1 an converges iff M f (x)dx is convergent.
gence is the set of x values for which the series converges:
Theorem 8 P
(Comparison test). Consider two series
P
∞
∞
+
a
and
n=1 n
n=1 bn satisfying ∀n ∈ N , 0 < an ≤ bn .
P∞
P∞
• If n=1 bn converges, then n=1 an converges.
P∞
P∞
• If n=1 an diverges, then n=1 bn diverges.
Ic (p) = {x | p(x) converges}.
(7)
Theorem 17 (Interval of convergence of power series).
There are only three possibilities for Ic (p) of the power
series (6): (i) Ic = {a}, (ii) Ic = (−∞, +∞),
Theorem
9P
(Limit comparison test). If two series (iii) Ic = (a − R, a + R) or [a − R, a + R) or (a − R, a + R]
P∞
∞
+
n=1 an and
n=1 bn satisfy ∀n ∈ N , an > 0, bn > 0, or [a − R, a + R].
and
R is called the radius of convergence. R = 0, +∞ for
an
= c ∈ (0, ∞),
(3) cases (i) & (ii), respectively.
lim
n→∞ b
n
then either both series converge or both diverge.
Theorem 18 (Term-by-term
differentiation and integraP∞
tion). If f (x) = n=0 cn (x − a)n has radius of convergence R > 0, then f (x) is differentiable on (a − R, a + R).
Theorem 10 (AlternatingP
series test). An alternating
∞
series is a series of the form n=1 (−1)n−1 bn with bn > 0
∀n ∈ N+ . If it satisfies
(i) ∀n ≥ 1, bn+1 ≤ bn , and (ii)
P∞
limn→∞ bn = 0, then n=1 (−1)n−1 bn converges.
P∞
Theorem 11P(Absolute convergence). If n=1 |an | con∞
verges, then n=1 an converges.
P∞
Theorem 12 (Ratio test). Consider a series Pn=1 an
∞
an+1
satisfying limn→∞ | an | = L. If L < 1, then n=1 an
P∞
converges; if L > 1, then n=1 an diverges; if L = 1, no
conclusion can be drawn.
f 0 (x) =
∞
X
ncn (x − a)n−1 ,
(8)
n=0
Z
f (x)dx = C +
∞
X
n=0
cn
(x − a)n+1
.
n+1
(9)
Furthermore,
the radii of convergence of both f 0 (x) and
R
f (x)dx are R.
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Math 1321-004 (Qinghai Zhang)
5
Memorization sheet for Exam 1
Taylor Series
Theorem 23 (Taylor’s inequality). If ∃M < ∞, s.t.
∀x ∈ [a−d, a+d], f (n+1) (x) ≤ M , then ∀x ∈ (a−d, a+d),
Definition 19. If f (n) (x) exists for a function f : R → R
at x = a, then
Tn (x) =
n
X
k=0
f (k) (a)
(x − a)k
k!
|Rn (x)| ≤
n→∞
∞
X
f (k) (a)
k=0
k!
(x − a)k
(15)
Axiom 24 (Fundamental principle of counting). A task
consists of a sequence of k steps. Let ni denote the number of different choices for the i-th step, the total number
of distinct ways to complete the task is then
(11)
k
Y
is called the Taylor series for f (x) at a. The remainder
of the nth Taylor polynomial in approximating f (x) is
Rn (x) = f (x) − Tn (x).
M
|x − a|n+1 .
(n + 1)!
(10) If ∀n ∈ N, (15) holds, then limn→∞ Tn = f (x).
is called the nth Taylor polynomial for f (x) at a.
lim Tn (x) =
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ni = n1 n2 · · · nk .
(16)
i=1
Definition 25. ∀k, n ∈ N, the binomial coefficients are
( k!
if k ≥ n,
k
(17)
= n!(k−n)!
n
0
if k < n.
(12)
(m)
Lemma 20. ∀m = 0, 1, 2, . . . , n, Rn (a) = 0.
∀k ∈ R, ∀n ∈ N, the binomial coefficients are
( Qn−1 (k−i)
i=0
k
if n > 0,
n!
=
(13)
n
1
if n = 0.
Theorem 21. Let Tn be the nth Taylor polynomial for
f (x) at a.
lim Rn (x) = 0 ⇔ f (x) = lim Tn (x).
n→∞
n→∞
(18)
Theorem 22 (Taylor’s theorem with Lagrangian form). Theorem 26 (Binomial series). ∀k ∈ R, |x| < 1,
Consider a function f : R → R. If f (n+1) (x) exists on
∞ Qn−1
∞ X
X
k n
the interval I = (a − d, a + d), and f (n) (x) is continuous
i=0 (k − i) n
x =1+
x (19)
(1 + x)k =
on [a − d, a + d], then ∀x ∈ I, ∃y ∈ I s.t.
n
n!
n=0
Rn (x) =
f (n+1) (y)
(x − a)n+1 .
(n + 1)!
n=1
∞
X
k(k − 1) · · · (k − n + 1) n
=1+
x
n!
n=1
(14)
2
(20)