- •Functions
- •The Concept of a Function
- •Trigonometric Functions
- •Inverse Trigonometric Functions
- •Logarithmic, Exponential and Hyperbolic Functions
- •Limits and Continuity
- •Introductory Examples
- •Continuity Examples
- •Linear Function Approximations
- •Limits and Sequences
- •Properties of Continuous Functions
- •The Derivative
- •The Chain Rule
- •Higher Order Derivatives
- •Mathematical Applications
- •Antidifferentiation
- •Linear Second Order Homogeneous Differential Equations
- •Linear Non-Homogeneous Second Order Differential Equations
- •Area Approximation
- •Integration by Substitution
- •Integration by Parts
- •Logarithmic, Exponential and Hyperbolic Functions
- •The Riemann Integral
- •Volumes of Revolution
- •Arc Length and Surface Area
- •Techniques of Integration
- •Integration by formulae
- •Integration by Substitution
- •Integration by Parts
- •Trigonometric Integrals
- •Trigonometric Substitutions
- •Integration by Partial Fractions
- •Fractional Power Substitutions
- •Numerical Integration
- •Integrals over Unbounded Intervals
- •Discontinuities at End Points
- •Improper Integrals
- •Sequences
- •Monotone Sequences
- •Infinite Series
- •Series with Positive Terms
- •Alternating Series
- •Power Series
- •Taylor Polynomials and Series
- •Applications
- •Parabola
- •Ellipse
- •Hyperbola
- •Polar Coordinates
- •Graphs in Polar Coordinates
- •Areas in Polar Coordinates
- •Parametric Equations
320 |
CHAPTER 8. INFINITE SERIES |
8.2Monotone Sequences
Definition 8.2.1 Let {tn}∞n=m be a given sequence. Then {tn}∞n=m is said to be
(a)increasing if tn < tn+1 for all n ≥ m;
(b)decreasing if tn+1 < tn for all n ≥ m;
(c)nondecreasing if tn ≤ tn+1 for all n ≥ m;
(d)nonincreasing if tn+1 ≤ tn for all n ≥ m;
(e)bounded if a ≤ tn ≤ b for some constants a and b and all n ≥ m;
(f)monotone if {tn}∞n=m is increasing, decreasing, nondecreasing or nonincreasing.
(g)a Cauchy sequence if for each > 0 there exists some M such that |an1 − an2 | < whenever n1 ≥ M and n2 ≥ M.
Theorem 8.2.1 (a) A monotone sequence converges to some real number if and only if it is a bounded sequence.
(b) A sequence is convergent if and only if it is a Cauchy sequence.
Proof.
Part (a) Suppose that an ≤ an+1 ≤ B for all n ≥ M and some B. Let L be the least upper bound of the sequence {an}∞n=m. Let > 0 be given. Then there exists some natural number N such that
L − < aN ≤ L.
Then for each n ≥ N, we have
L − < aN ≤ an ≤ L.
By definition {an}∞n=m converges to L.
Similarly, suppose that B ≤ an+1 ≤ an for all n ≥ M. Let L be the greatest lower bound of {an}∞n=m. Then {an}∞n=m converges to L. It follows that a bounded monotone sequence converges. Conversely, suppose that a
8.2. MONOTONE SEQUENCES |
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monotone sequence {an}∞n=m converges to L. Let = 1. Then there exists some natural number N such that if n ≥ N, then
|an − L| < − < an − L < L − < an < L + .
The set {an : m ≤ n ≤ N} is bounded and the set {an : n ≥ N} is bounded. It follows that {an}∞n=m is bounded. This completes the proof of Part (a) of the theorem.
Part (b) First, let us suppose that {an}∞n=m converges to L. Let > 0 be given. Then 2 > 0 and hence there exists some natural number N such that for all natural numbers p ≥ N and q ≥ N, we have
|ap − L| < |
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|ap − aq| = |(ap − L) + (L + aq)|
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= .
It follows that {an}∞n=m is a Cauchy sequence.
Next, we suppose that {an}∞n=m is a Cauchy sequence. Let S = {an : m ≤ n < ∞}. Suppose > 0. Then there exists some natural number N such
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It follows that S is a bounded set. If S is an infinite set, then S has some limit point q and some subsequence {ank }∞k=1 of {an}∞n=m that converges to q. Since > 0, there exists some natural number M such that for all k ≥ M, we have
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322 CHAPTER 8. INFINITE SERIES
Also, for all k ≥ N + M, we get nk ≥ k ≥ N + M and
|ak − q| = |ak − ank + ank − q| ≤ |ank − ak| + |ank − q|
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It follows that the sequence {an}∞n=m converges to q. If S is a finite set, then some ak is repeated infinite number of times and hence some subsequences of {an}∞n=m converges to ak. By the preceding argument {an}∞n=m also converges to ak. This completes the proof of this theorem.
Theorem 8.2.2 Let {f(n)}∞n=m be a sequence where f is a di erentiable function defined for all real numbers x ≥ m. Then the sequence {f(n)}∞n=m is
(a)increasing if f0(x) > 0 for all x > m;
(b)decreasing if f0(x) < 0 for all x > m;
(c)nondecreasing if f0(x) ≥ 0 for all x > m;
(d)nonincreasing if f0(x) ≤ 0 for all x > m.
Proof. Suppose that m ≤ a < b. Then by the Mean Value Theorem for derivatives, there exists some c such that a < c < b and
f(b) − f(a) = f0(c), b − a
f(b) = f(a) + f0(c)(b − a).
The theorem follows from the above equation by considering the value of f0(c). In particular, for all natural numbers n ≥ m,
f(n + 1) = f(n) + f0(c),
for some c such that n < c < n + 1.
Part (a). If f0(c) > 0, then f(n + 1) > f(n) for all n ≥ m. Part (b). If f0(c) < 0, then f(n + 1) < f(n) for all n ≥ m. Part (c). If f0(c) ≥ 0, then f(n + 1) ≥ f(n) for all n ≥ m. Part (d). If f0(c) ≤ 0, then f(n + 1) ≤ f(n) for all n ≤ m.
This completes the proof of this theorem.
8.3. INFINITE SERIES |
323 |
8.3Infinite Series
Definition 8.3.1 Let {tn}∞n=1 be a given sequence. Let
n
X s1 = t1, s2 = t1 + t2, s3 = t1 + t2 + t3, · · · , sn = tk,
k=1
for all natural number n. If the sequence {sn}∞n=1 converges to a finite number
L, then we write
∞
X
L = t1 + t2 + t3 + · · · = tk.
k=1
n
X
We call tk an infinite series and write
k=1
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tk = lim |
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n→∞ |
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We say that L is the sum of the series and the series converges to L. If a series does not converge to a finite number, we say that it diverges. The sequence {sn}∞n=1 is called the sequence of the nth partial sums of the series.
Theorem 8.3.1 Suppose that a and r are real numbers and a 6= 0. Then the geometric series
∞
a + ar + ar2 + · · · = Xark = 1 −a r,
k=0
if |r| < 1. The geometric series diverges if |r| ≥ 1.
Proof. For each natural number n, let
sn = a + ar + · · · + arn−1.
On multiplying both sides by r, we get
rsn = ar + ar2 + · · · + arn−1 + arn
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The rest of the theorem follows from the preceding argument. This completes the proof of this theorem.
Theorem 8.3.3 (The Integral Test) Let f be a function that is defined, continuous and decreasing on [1, ∞) such that f(x) > 0 for all x ≥ 1. Then
∞Z ∞
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Proof. Suppose that f is decreasing and continuous on [1, ∞), and f(x) > 0 for all x ≥ 1. Then for all natural numbers n, we get,
n+1 f(k) ≤ Z n+1 f(x)dx ≤ |
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8.3. INFINITE SERIES |
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graph
It follows that,
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Since f(1) is a finite number, it follows that
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Theorem 8.3.4 Suppose that p > 0. Then the p-series
∞
X 1
np
n=1
converges if p > 1 and diverges if
series P∞ 1 diverges.
n=1 n
0 < p ≤ 1. In particular, the harmonic
Proof. Suppose that p > 0. Then
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326 |
CHAPTER 8. INFINITE SERIES |
Exercises 8.1
1.Define the statement that the sequence {an}∞n=1 converges to L.
2.Suppose the sequence {an}∞n=1 converges to L and the sequences {bn}∞n=1 converges to M. Then prove that
(a){can}∞n=1 converges to cL, where c is constant.
(b){an + bn}∞n=1 converges to L + M.
(c){an − bn}∞n=1 converges to L − M.
(d){anbn}∞n=1 converges to LM.
∞
(e)an converges to L , if M 6= 0. bn n=1 M
3.Suppose that 0 < an ≤ an+1 < M for each natural number n. Then prove that
(a){an}∞n=1 converges.
(b){−an}∞n=1 converges.
(c)akn ∞n=1 converges for each natural number k.
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