Calculus 3 3: chuỗi hội tụ

Infinite Series and Differential Equations Nguyen Thieu Huy Hanoi University of Science and Technology Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 14 Calculation of radius of convergence Theorem For an x n with ρ := lim radius of convergence is R Nguyen Thieu Huy (HUST) an+1 (or ρ := lim n |an |) we have that the an n→∞ = ρ1 with the conventions 10 = ∞ and ∞ = n→∞ Infinite Series and Diff Eq / 14 Calculation of radius of convergence Theorem an x n with ρ := lim For an+1 (or ρ := lim n |an |) we have that the an n→∞ = ρ1 with the conventions 10 = ∞ and ∞ = n→∞ radius of convergence is R Note: an x n is absolutely convergent ∀x ∈ (−R, R) To calculate the domain of conv for as previously, or Nguyen Thieu Huy (HUST) an x n we can either compute Infinite Series and Diff Eq / 14 Calculation of radius of convergence Theorem an x n with ρ := lim For an+1 (or ρ := lim n |an |) we have that the an n→∞ = ρ1 with the conventions 10 = ∞ and ∞ = n→∞ radius of convergence is R Note: an x n is absolutely convergent ∀x ∈ (−R, R) To calculate the domain of conv for an x n we can either compute as previously, or 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 / 14 Calculation of radius of convergence Theorem an x n with ρ := lim For an+1 (or ρ := lim n |an |) we have that the an n→∞ = ρ1 with the conventions 10 = ∞ and ∞ = n→∞ radius of convergence is R Note: an x n is absolutely convergent ∀x ∈ (−R, R) To calculate the domain of conv for an x n we can either compute as previously, or 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 n X = f (x) and reducing it to the power series an X Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 14 Properties of power series Uniform convergence Let power series an x n have radius of convergence R > Then, an x n converges uniformly in any closed interval which lies entirely within its interval of convergence (−R, R) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 14 Properties of power series Uniform convergence Let power series an x n have radius of convergence R > Then, an x n converges uniformly in any closed interval which lies entirely within its interval of convergence (−R, R) Proof Fix any [a, b] ⊂ (−R, R), then −R < a < b < R So, ∃x0 ∈ (0, R) such that [a, b] ⊂ [−x0 , x0 ] We have power series converges absolutely at x0 Moreover, |an x n | |an x0n | ∀x ∈ [a, b], ∀n, and |an x0n | is convergent Thanks to Weierstrass, power series converges uniformly on [a, b] Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 14 Properties of power series Uniform convergence Let power series an x n have radius of convergence R > Then, an x n converges uniformly in any closed interval which lies entirely within its interval of convergence (−R, R) Proof Fix any [a, b] ⊂ (−R, R), then −R < a < b < R So, ∃x0 ∈ (0, R) such that [a, b] ⊂ [−x0 , x0 ] We have power series converges absolutely at x0 Moreover, |an x n | |an x0n | ∀x ∈ [a, b], ∀n, and |an x0n | is convergent Thanks to Weierstrass, power series converges uniformly on [a, b] Corollary (Continuity) Power series is continuous on any closed interval lying entirely within its interval of convergence, therefore it is continuous on interval of ∞ convergence, and lim x→x0 n=0 Nguyen Thieu Huy (HUST) an x n = ∞ n n=0 an x0 Infinite Series and Diff Eq ∀x ∈ (−R, R) / 14 Corollary (Integrability) Power series is integrable on any closed interval lying entirely within its interval of convergence, and b ∞ a n=0 an x n dx = ∞ b n=0 a an x n dx = ∞ n=0 an n+1 n+1 (b − an+1 ) ∀[a, b] ⊂ (−R, R) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 14 Corollary (Integrability) Power series is integrable on any closed interval lying entirely within its interval of convergence, and b ∞ a n=0 an x n dx = ∞ b ∞ an x n dx = n=0 n=0 a an n+1 n+1 (b − an+1 ) ∀[a, b] ⊂ (−R, R) Corollary (Differentiability) Power series is infinitely many times differentiable on (−R, R), and dk dx k ∞ n=0 an x n ∞ = n=0 dk a xn dx k n ∞ = n(n − 1) · · · (n − k + 1)an x n−k for all n=k k ∈ N Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 14 Example Example 1: Expand to Maclaurin’s Series for f (x) = sin x ∀x ∈ R f (x) = cos x; f (x) = − sin x; f (x) = − cos x; f (4) (x) = sin x; We have f (n) (x) = (sin x)(n) = sin(x + nπ ) ∀x ∈ R |f (n) (x)| ∀x ∈ R, ∀n ∈ N ∞ ∞ sin( nπ ) n n! x n=0 n=0 ∞ (−1)k x 2k Similarly, cos x = (2k)! k=0 ∞ k x Easy to see e x = k! k=0 Then, sin x = Example 2: Example 3: sin(n) (0) n x n! = ∞ = k=0 (−1)k x 2k+1 (2k+1)! Application: Euler’s formula: substituting x = iϕ we obtain e iϕ = ∞ k=0 (iϕ)k k! ∞ = n=0 (iϕ)2n (2n)! ∞ + n=0 (iϕ)2n+1 (2n+1)! ∞ = n=0 (−1)n ϕ2n (2n)! ∞ +i n=0 (−1)n ϕ2n+1 (2n+1)! = cos ϕ + i sin ϕ Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 14 SOME IMPORTANT POWER SERIES 1−x = + x + x2 + · · · + xn + · · · Nguyen Thieu Huy (HUST) − < x < −1 Infinite Series and Diff Eq / 14 Fourier Series Periodic functions: A function f (x) is said to be periodic with period T if for all x ∈ R, f (x + T ) = f (x), where T is a positive constant The least value of T > is called the least period or simply the period of f (x) Examples: Functions sin x, cos x are periodic with period T = 2π, πx whereas, cos πx L and sin L are periodic with period 2L Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 10 / 14 Fourier Series Periodic functions: A function f (x) is said to be periodic with period T if for all x ∈ R, f (x + T ) = f (x), where T is a positive constant The least value of T > is called the least period or simply the period of f (x) Examples: Functions sin x, cos x are periodic with period T = 2π, πx whereas, cos πx L and sin L are periodic with period 2L A periodic function with period T is called a T -periodic function Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 10 / 14 Piecewise continuous functions Definition A function f (x) is said to be piecewise continuous on (a, b) if (a, b) can be subdivided into a finite number of subinterval a = x0 < x1 < · · · < xn = b such that on each open subinterval (xj , xj+1 ) the function is continuous ∀j = 0, n − and at each endpoint xj the limits from the right and from the left exist We denote limit from the right and limit from the left Nguyen Thieu Huy (HUST) lim lim x→xj ; x>xj x→xj ; x 0: using formulas Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 13 / 14 For m = 0: L f L (x)dx = −L a0 dx = La0 , thus a0 = L L f (x)dx −L For m > 0: using formulas L we have −L f (x) cos mπx L dx = Lam Therefore, am = L L −L f (x) cos mπx L dx for all m = 1, 2, · · · Combining with the case m = 0, we obtain am = L L −L f (x) cos mπx L dx for all m = 0, 1, 2, · · · Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 13 / 14 For m = 0: L f L (x)dx = −L a0 dx = La0 , thus a0 = L L f (x)dx −L For m > 0: using formulas L we have −L f (x) cos mπx L dx = Lam Therefore, am = L L −L f (x) cos mπx L dx for all m = 1, 2, · · · Combining with the case m = 0, we obtain am = bm = L L L −L L −L f (x) cos mπx L dx for all m = 0, 1, 2, · · · Similarly, we have f (x) sin mπx L dx for all m = 1, 2, · · · Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 13 / 14 For m = 0: L f L (x)dx = −L a0 dx = La0 , thus a0 = L L f (x)dx −L For m > 0: using formulas L we have −L f (x) cos mπx L dx = Lam Therefore, am = L L −L f (x) cos mπx L dx for all m = 1, 2, · · · Combining with the case m = 0, we obtain am = bm = L L L −L L −L f (x) cos mπx L dx for all m = 0, 1, 2, · · · Similarly, we have f (x) sin mπx L dx for all m = 1, 2, · · · Therefore, if such a series exists, then it is unique (because all an , bn are uniquely determined by f ) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 13 / 14 Definition Let f (x) be defined on R, periodic with period 2L, and piecewise continuous on (−L, L) Then, the series of trigonometric functions ∞ a0 nπx + an cos nπx in which L + bn sin L n=1 an = bn = L L L −L L −L f (x) cos nπx L dx for all n = 0, 1, 2, · · · f (x) sin nπx L dx for all n = 1, 2, · · · is called the Fourier Series of f Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 14 / 14 Definition Let f (x) be defined on R, periodic with period 2L, and piecewise continuous on (−L, L) Then, the series of trigonometric functions ∞ a0 nπx + an cos nπx in which L + bn sin L n=1 an = bn = L L L −L L −L f (x) cos nπx L dx for all n = 0, 1, 2, · · · f (x) sin nπx L dx for all n = 1, 2, · · · is called the Fourier Series of f The above-defined numbers an , bn are called Fourier Coefficients of f Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 14 / 14