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Sequences and Series of Functions Sequences and Series of Functions Li Jiayi, Joanna 14 July, 2014 1 Contents 1 Review: Sequences and Series of Real Numbers 4 1.1 Sequences . .4 1.1.1 Sequences . .4 1.1.2 Some properties of a convergent sequence . .4 1.1.3 Monotone Sequences . .5 1.1.4 Subsequences and the Bolzano-Weiertrass Theorem . .6 1.1.5 Cauchy’s Convergence Criterion . .7 1.2 Series . .7 1.2.1 Series . .7 1.2.2 Some tests for convergence . .8 1.2.3 Absolute and Conditional Convergence . .8 2 Sequences and Series of Functions 9 2.1 Pointwise convergence of sequences of functions . .9 2.2 Uniform convergence of sequences of functions . 11 2.2.1 Uniform convergence and boundedness . 13 2.2.2 Uniform convergence and continuity . 14 2.2.3 Uniform convergence and Riemann integration . 16 2.2.4 Uniform convergence and differentiation . 17 2 2.3 Uniform convergence of series of functions . 20 2.3.1 Some tests for uniform convergence of series of functions . 20 2.3.2 the Power Series . 20 3 1 Review: Sequences and Series of Real Numbers 1.1 Sequences 1.1.1 Sequences Definition 1.1. A sequence of real numbers is a real-valued function defined on the set of natural numbers, i.e. a function f : N −! R. In other words, a sequence can be written as f(1); f(2); f(3); ::: We shall denote by an such a sequence where an = f(n) 2 R: The number an is called the n-th term of the sequence. Definition 1.2. A sequence fang is said to converge to a real number L if 8 > 0; 9N 2 N such that if n > N, then jan − Lj < . n Example 1.1. Prove that the sequence converges to 1, that is, prove that n + 1 n lim = 1 x!1 n + 1 Proof. We are required to show that for any given > 0, we can find a natural number N n −1 such that if n > N, then j − 1j = j j < . n + 1 n + 1 1 1 We first note that < for any N 2 . Now that for any given , by the Archimedean n + 1 n N 1 Property, there exists N 2 such that < . It follows that if n > N, then N N 1 1 1 < < < n + 1 n N and so we are done. 1.1.2 Some properties of a convergent sequence Theorem 1.1 (Uniqueness of Limits). Let fang be a convergent sequence. Then the limit of the sequence is unique. 4 Definition 1.3. A sequence fang is bounded above if there exists M 2 R such that an < M for all n 2 N. Similarly, a sequence fang is bounded below if there exists m 2 R such that an > m for all n 2 N. A sequence fang is bounded if it is bounded above and below. Theorem 1.2 (Boundedness of Convergent Sequences). If fang is a convergent sequence, then it is bounded. Theorem 1.3 (Arithmetic Properties of Convergent Sequences)). Let fang and fbng be two sequences. Suppose lim an = A and lim bn = B for some A, B 2 . Then n!1 n!1 R (a) lim (an + bn) = lim an + lim bn = A + B n!1 n!1 n!1 (b) lim (anbn) = ( lim an)( lim bn) = AB n!1 n!1 n!1 (c) lim can = c lim an = cA for any c 2 . n!1 n!1 R an limn!1 an A (d) lim = = , provided that bn 6= 0 for all n 2 N and B 6= 0. n!1 bn limn!1 bn B Theorem 1.4 (Sandwich Theorem). Let fang; fbng and fcng be sequences. Suppose that an ≤ bn ≤ cn for all n 2 , and that lim an = lim cn = L. Then lim bn = L. N n!1 n!1 n!1 1.1.3 Monotone Sequences Definition 1.4. Let fang be a sequence. We say that fang is increasing if an ≤ an+1 for all n 2 N and is decreasing if an ≥ an+1 for all n 2 N. We say that fang is a monotone if it is either increasing or decreasing. Theorem 1.5. (a) If fang is increasing and bounded above, then fang converges. (b) If fang is decreasing and bounded below, then fang converges. 5 1.1.4 Subsequences and the Bolzano-Weiertrass Theorem Definition 1.5. Let fang be a sequence of real numbers, and let n1 < n2 < n3 < ··· be a strictly increasing sequence of natural numbers. Then the sequence an1 ; an2 ; an3 ··· is called a subsequence of fang and is denoted by fank g, where k 2 N indexes the subse- quence. Theorem 1.6. If fang is a sequence converging to L, then every subsequence fank g also converges to L. Theorem 1.7. Let fang be a sequence. The fang has a monotone subsequence. , Proof. To prove this theorem, we call the k-th term dominant if it is greater than or equal to all the following terms. In other words, the term ak is dominant if ak ≥ am for all m ≥ k: There are two cases to consider: Case 1: There are infinitely many dominant terms in the sequence fang: Then we have a infinite subsequence an1 ; an2 ; an3 ;::: Now ani is dominant for all i 2 N and ni < ni+1, so we must haveani ≥ ani+1 . Similarly, we have an1 ≥ an2 ≥ an3 ≥ · · · Hence we have a decreasing subsequence ank and we are done. Case 2: There are only finitely many dominant terms in the sequence fang: (including the case where there is no dominant term). Suppose aN is the last dominant term, and let n1 = N + 1. Now an1 is not dominant and so there must exist n2 > n1 such that an1 ≤ an2 . Continuing this way, we obtain an increasing subsequence an1 ; an2 ; an3 ; ··· as desired. Theorem 1.8 (the Bolzano-Weiertrass Theorem). Let fang be a bounded sequence. Then fang has a convergent subsequence. 6 1.1.5 Cauchy’s Convergence Criterion Definition 1.6. A sequence fang is called a Cauchy sequence if 8 > 0; 9N 2 N such that if n; m > N, then jan − amj < . Theorem 1.9. Let fang be a convergent sequence. Then fang is a Cauchy sequence. Theorem 1.10. Let fang be a Cauchy sequence. Then fang is a convergent sequence. , 1.2 Series 1.2.1 Series Definition 1.7. Let fang be a sequence of real numbers. The sequence fskg can be defined by s1 = a1 s2 = a1 + a2 s3 = a1 + a2 + a3 . k X sk = a1 + a2 + a3 + ··· + ak = an n=1 1 X is called the sequence of partial sums of the series an. n=1 The term an is called the n-th term of the series and the term sk is called the k-th partial 1 X sum of the series. The series an is said to converge to some number L if the associated n=1 sequence fskg of the partial sums converges to L. We write 1 X an = L n=1 7 and we refer to L as the sum of the series. If the sequence fskg of partial sums diverges, then 1 X we say that the series an diverges. n=1 1.2.2 Some tests for convergence 1 X Theorem 1.11 (Cauchy Convergence Criterion for Series). The series an converges if n=1 and only if for any > 0 there exists N 2 N such that if n > m > N, then jam+1 + am+2 + ··· + anj < . Theorem 1.12 (Comparison Test). Suppose that fang and fbng are sequences satisfying 0 ≤ an ≤ bn for all n 2 N. 1 1 X X (a) If bn, then anconverges n=1 n=1 1 1 X X (b) If an diverges, then bn diverges. n=1 n=1 Theorem 1.13 (Ratio Test). Let fang be a sequence with an > 0 for all n 2 N. Suppose that a lim n n!1 an+1 1 X exists and equals r. The the series an converges if r < 1 and diverges if r > 1. n=1 1.2.3 Absolute and Conditional Convergence 1 X Definition 1.8. Let fang be a sequence of real numbers. The series an is said to be n=1 1 1 X X absolutely convergent if the series janj is convergent. The series an is said to be n=1 n=1 1 X conditionally convergent if it converges but does not converge absolutely, that is, an con- n=1 1 X verges, but janj diverges. n=1 8 2 Sequences and Series of Functions We will now discuss the convergence of sequences and series of real-valued functions de- fined on a set X. In most cases, X would be a subset of R. Since we are dealing with the convergence problems of sequences and series, we will naturally be interested in knowing if the limit functions, whenever they exist, will preserve properties such as boundedness, con- tinuity, uniform continuity, differentiability and integrability. In other words, if the sequence of functions under consideration is bounded, continuous, differentiable or integrable, and if the limit function of the sequence exists, does it carry these properties? 2.1 Pointwise convergence of sequences of functions Definition 2.1. Suppose ffng, n = 1; 2; 3 ··· is a sequence of functions defined on a set E, and suppose that the sequence of numbers ffn(x)g converges for every x 2 E. We can define a function f by f(x) = lim fn(x); x 2 E: n!1 Under these circumstances, we say that ffng converges on E and that f is the limit, or the limitfunction, of ffng.
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