Infinite Series and Differential Equations Nguyen Thieu Huy Hanoi University of Science and Technology Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 Series of Functions Definition Let {un (x)}∞ n=1 be a sequence of functions defined ∀x ∈ D ⊂ R Formal sum ∞ un (x) is called a series of functions n=1 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 Series of Functions Definition Let {un (x)}∞ n=1 be a sequence of functions defined ∀x ∈ D ⊂ R Formal sum ∞ un (x) is called a series of functions n=1 ∞ Here, when x is taken a concrete real value x0 ∈ D then un (x0 ) is a n=1 series of numbers Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 Series of Functions Definition Let {un (x)}∞ n=1 be a sequence of functions defined ∀x ∈ D ⊂ R Formal sum ∞ un (x) is called a series of functions n=1 ∞ Here, when x is taken a concrete real value x0 ∈ D then un (x0 ) is a n=1 series of numbers ∞ n (x+1)n Ex 1: , ∀x ∈ R, is a series of functions Substituting x by 3, n! n=1 ∞ we obtain n=1 8n n! is a series of numbers Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 Series of Functions Definition Let {un (x)}∞ n=1 be a sequence of functions defined ∀x ∈ D ⊂ R Formal sum ∞ un (x) is called a series of functions n=1 ∞ Here, when x is taken a concrete real value x0 ∈ D then un (x0 ) is a n=1 series of numbers ∞ n (x+1)n Ex 1: , ∀x ∈ R, is a series of functions Substituting x by 3, n! n=1 ∞ we obtain n=1 8n n! is a series of numbers ∞ Ex 2: Series of functions ∞ n=1 n=1 √1 n nx , ∀x ∈ R Substituting x by 12 , we have is a series of numbers Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 Domain of Convergence Definition ∞ un (x), the set X ⊂ R is called ∞   un (x) is convergent ∀x ∈ X  n=1 domain of convergence ⇐⇒ ∞   un (x) is divergent ∀x ∈ / X  For series of functions n=1 n=1 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 Domain of Convergence Definition ∞ un (x), the set X ⊂ R is called ∞   un (x) is convergent ∀x ∈ X  n=1 domain of convergence ⇐⇒ ∞   un (x) is divergent ∀x ∈ / X  For series of functions n=1 n=1 ∞ In that case, for each x ∈ X we put S(x) := un (x) ∈ R, then we n=1 obtain a function x → S(x) The function S(x) is called the sum of the ∞ series of functions ∞ un (x), and we say n=1 un (x) is convergent on X to n=1 the function S(x) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 Domain of Convergence Definition ∞ un (x), the set X ⊂ R is called ∞   un (x) is convergent ∀x ∈ X  n=1 domain of convergence ⇐⇒ ∞   un (x) is divergent ∀x ∈ / X  For series of functions n=1 n=1 ∞ In that case, for each x ∈ X we put S(x) := un (x) ∈ R, then we n=1 obtain a function x → S(x) The function S(x) is called the sum of the ∞ series of functions ∞ un (x), and we say n=1 un (x) is convergent on X to n=1 the function S(x) ∞ Ex 1: Domain of convergence for 1−x ∞ = x n is (−1, 1) Its sum is n=0 x n; ∀x ∈ (−1, 1) n=0 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 ∞ Ex 2: Domain of convergence for ∞ Its sum S(x) := n=1 Nguyen Thieu Huy (HUST) n=1 nx nx is (1, ∞) (property of p-Series) is called ζ-Riemann function Infinite Series and Diff Eq / 15 ∞ Ex 2: Domain of convergence for ∞ Its sum S(x) := n=1 n=1 nx nx is (1, ∞) (property of p-Series) is called ζ-Riemann function ♣ The previous tests can be applied to find domains of convergence Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 15 ? Checking = From the facts that xn n ∞ is cont diff ∀n, n=1 xn n ∞ xn is conv to S(x) and n=1 ? is uni conv on [− 12 , 12 ], we have that S(x) is cont diff., and = holds true Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 12 / 15 ? Checking = From the facts that xn n ∞ is cont diff ∀n, n=1 xn n ∞ xn is conv to S(x) and n=1 ? is uni conv on [− 12 , 12 ], we have that S(x) is cont diff., and = holds true III Power Series Definition: A series of functions having the form ∞ a0 + a1 x + a2 x + · · · = an x n ; (convention: x = ∀x) n=0 where a0 , a1 , a2 , · · · , are real constants, is called a power series in x We abbreviate the above series as an x n Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 12 / 15 ? Checking = From the facts that xn n ∞ is cont diff ∀n, n=1 xn n ∞ xn is conv to S(x) and n=1 ? is uni conv on [− 12 , 12 ], we have that S(x) is cont diff., and = holds true III Power Series Definition: A series of functions having the form ∞ a0 + a1 x + a2 x + · · · = an x n ; (convention: x = ∀x) n=0 where a0 , a1 , a2 , · · · , are real constants, is called a power series in x We abbreviate the above series as an x n ∞ Ex.: n=1 xn n , ∞ n=0 3n x n n! Nguyen Thieu Huy (HUST) are power series Infinite Series and Diff Eq 12 / 15 Abel’s Theorem Theorem (Abel) If an x n converges at the point x0 = 0, then it converges absolutely at any point x satisfying |x| < |x0 | If an x n diverges at the point x1 , then it diverges at any point x satisfying |x| > |x1 | Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 13 / 15 Abel’s Theorem Theorem (Abel) If an x n converges at the point x0 = 0, then it converges absolutely at any point x satisfying |x| < |x0 | If an x n diverges at the point x1 , then it diverges at any point x satisfying |x| > |x1 | PROOF (1) We have an x0n converges Now, fix any x ∈ R with |x| < |x0 |, and compute |an x n | = |an x0n | Nguyen Thieu Huy (HUST) x x0 n Infinite Series and Diff Eq 13 / 15 Abel’s Theorem Theorem (Abel) If an x n converges at the point x0 = 0, then it converges absolutely at any point x satisfying |x| < |x0 | If an x n diverges at the point x1 , then it diverges at any point x satisfying |x| > |x1 | PROOF (1) We have an x0n converges Now, fix any x ∈ R with n |x| < |x0 |, and compute |an x n | = |an x0n | xx0 Since, the series an x0n converges, we have that limn→∞ an x0n = So, the sequence {an x0n } is bounded Therefore, ∃M > 0, such that |an x0n | M ∀n Thus, |an x n | M x x0 n ∀n n Since |x| < |x0 |, we have M xx0 is conv The assertion now follows from the comparison test (2) is consequence of (1) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 13 / 15 Radius and Interval of of Convergence Definition For an x n , the number R ∞ is called the Radius of Convergence an x n is convergent ∀|x| < R ⇐⇒ an x n is divergent ∀|x| > R Then, the interval (−R, R) is called the interval of convergence Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 14 / 15 Radius and Interval of of Convergence Definition For an x n , the number R ∞ is called the Radius of Convergence an x n is convergent ∀|x| < R ⇐⇒ an x n is divergent ∀|x| > R Then, the interval (−R, R) is called the interval of convergence Rem.: Two special cases: R = ⇐⇒ an x n is convergent at only point x = R = ∞ ⇐⇒ an x n is convergent for all x ∈ R Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 14 / 15 Radius and Interval of of Convergence Definition For an x n , the number R ∞ is called the Radius of Convergence an x n is convergent ∀|x| < R ⇐⇒ an x n is divergent ∀|x| > R Then, the interval (−R, R) is called the interval of convergence Rem.: Two special cases: R = ⇐⇒ an x n is convergent at only point x = R = ∞ ⇐⇒ an x n is convergent for all x ∈ R Ex.: xn (1) n has radius of convergence R = since it is conv ∀|x| < 1, and div ∀|x| > Interval of conv is (−1, 1) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 14 / 15 Radius and Interval of of Convergence Definition For an x n , the number R ∞ is called the Radius of Convergence an x n is convergent ∀|x| < R ⇐⇒ an x n is divergent ∀|x| > R Then, the interval (−R, R) is called the interval of convergence Rem.: Two special cases: R = ⇐⇒ an x n is convergent at only point x = R = ∞ ⇐⇒ an x n is convergent for all x ∈ R Ex.: xn (1) n has radius of convergence R = since it is conv ∀|x| < 1, and div ∀|x| > Interval of conv is (−1, 1) xn (2) n! has radius of convergence R = ∞ since it is conv ∀x ∈ R Interval of conv is (−∞, ∞) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 14 / 15 Radius and Interval of of Convergence Definition For an x n , the number R ∞ is called the Radius of Convergence an x n is convergent ∀|x| < R ⇐⇒ an x n is divergent ∀|x| > R Then, the interval (−R, R) is called the interval of convergence Rem.: Two special cases: R = ⇐⇒ an x n is convergent at only point x = R = ∞ ⇐⇒ an x n is convergent for all x ∈ R Ex.: xn (1) n has radius of convergence R = since it is conv ∀|x| < 1, and div ∀|x| > Interval of conv is (−1, 1) xn (2) n! has radius of convergence R = ∞ since it is conv ∀x ∈ R Interval of conv is (−∞, ∞) (3) (n!)x n has radius of convergence R = since it is conv at only point x = Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 14 / 15 Calculation of radius of convergence Theorem an x n with ρ := lim an+1 an (or ρ := lim n |an |) we have that the n→∞ 1 if < ρ < ∞  ρ radius of convergence is R = if ρ = ∞   ∞ if ρ = 1 With the conventions = ∞ and ∞ = 0, we can write R = ρ1 For Nguyen Thieu Huy (HUST) n→∞ Infinite Series and Diff Eq 15 / 15 Calculation of radius of convergence Theorem an x n with ρ := lim an+1 an (or ρ := lim n |an |) we have that the n→∞ 1 if < ρ < ∞  ρ radius of convergence is R = if ρ = ∞   ∞ if ρ = 1 With the conventions = ∞ and ∞ = 0, we can write R = ρ1 For n→∞ Note on Domain of Conv.: To calculate the domain of conv for we can use one of the following two methods: To compute as previously, or Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq an x n 15 / 15 Calculation of radius of convergence Theorem an x n with ρ := lim an+1 an (or ρ := lim n |an |) we have that the n→∞ 1 if < ρ < ∞  ρ radius of convergence is R = if ρ = ∞   ∞ if ρ = 1 With the conventions = ∞ and ∞ = 0, we can write R = ρ1 For n→∞ Note on Domain of Conv.: To calculate the domain of conv for an x n we can use one of the following two methods: To compute as previously, or To compute the radius of conv R, and then interval of conv (−R, R) Then, check the two endpoints −R and R to decide whether they can be included in the domain of conv Outside the interval of conv (i.e., for |x| > R) we knew that the series is div Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 15 / 15 Calculation of radius of convergence Theorem an x n with ρ := lim an+1 an (or ρ := lim n |an |) we have that the n→∞ 1 if < ρ < ∞  ρ radius of convergence is R = if ρ = ∞   ∞ if ρ = 1 With the conventions = ∞ and ∞ = 0, we can write R = ρ1 For n→∞ Note on Domain of Conv.: To calculate the domain of conv for an x n we can use one of the following two methods: To compute as previously, or To compute the radius of conv R, and then interval of conv (−R, R) Then, check the two endpoints −R and R to decide whether they can be included in the domain of conv Outside the interval of conv (i.e., for |x| > R) we knew that the series is div The note can be applied to the series of the form an (f (x))n by putting X = f (x) and reducing it to the power series an X n Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 15 / 15