MATH 201 Chapter 11 Summary, Fall 2011

Chapter 11: Infinite Sequences and Series Section 1: Definition 1: A numerical sequence, or simply a sequence is a function a, N → R. Theorem 1: {an} is a sequence. We say that {an} converges to a limit L as n → ∞, and we write L an n = →∞ lim , if to each given ε > 0 , there corresponds a positive integer N such that < ε − ⇒ ≥ L a N n n Theorem 2: k k n = →∞ lim (Where k is a function). 0 lim 1 = ∞ n→ n Definition 2: {an} is a sequence. If an does not converge, we say that {an} diverge. Theorem 3: {an} and {bn} are sequence. If B b A a n n → → and the i. ( ) B A b a n n n ± = ± →∞ lim ii. AB a b n n n = →∞ lim iii. B A b a n n n = →∞ lim provided bn is nonzero and B is also nonzero. Theorem 4: The Sandwich Theorem: {an}, {bn}, {cn} are sequences L b L c a n c b a n n n n n n n n = ⇒    = = ∀ ≤ ≤ →∞ →∞ →∞ nlim lim lim Corollary to Sandwich Theorem (SCT): {an}, {cn} are sequences: 0 0 n → ⇒    → ∀ ≤ n n n n a c c a Theorem 5: {an} is a sequence, f(x) is a function: ( ) ( ) ( ) f L f a x f L a n n → ⇒    → continuous at L Theorem 6: {an} is a sequence, f(x) is a function: ( ) ( ) L a L x f n f n a n n n → ⇒    = ∀ = →∞ lim Theorem 7: Six important limits: 0 0 with lim ln α > = α →∞ n n n 1 lim = ∞ → n n n 0 1 with lim 1 > = →∞ x x n n 1 0 with lim < = →∞ x x n n IR x e n x x n n ∈  =      + →∞ with lim 1 IR x n x n n ∈ = →∞ 0 with ! lim Definition 3: {an} is a sequence. M ∈ IR . i. We say that an is decreasing if n a a n n ∀ + ≤ 1 ii. We say that an is bounded from above by M if n M an ∀ ≤ Theorem 8: Important Theorem (IT): }converges. { }bounded from above { }increasing { n n n a a a ⇒    }converges { }bounded from below { }decreasing { n n n a a a ⇒    }diverges. { }unbounded { n n a a ⇒ Section 2: Definition 1: {an} is a sequence. The expression .... 3 2 1 1 + + + = ∑ ∞ = a a a a n n is called an infinite series. The sequence of partial sum of ∑ ∞ n=1 n a is the sequence {Sn} defined by: 2 1 2 1 1 , a a S a S + = = n n a a a a S + + + + = .... 2 2 1 . We have:   = →∞ ∞ =∑ } diverges if { diverges } converges if { lim 1 n n n n n n S S S a Geometric series: 0 , , ≠ ∈ IR a a r ∑ ∞ = − + + + = 1 2 1 ... n n ar ar a ar is called a geometric series of ratio r. The sequence of partial sum    = ≠ − − = 1 if 1 if 1 1 r na r r r a S n n and    ≥ < − = ∑ ∞ = − 1 if diverges 1 if 1 1 1 x r r a ar n n Theorem 1: ∑ ∑ ∞ = ∞ = 1 1 and n n n n b a are series and k ∈ IR . If B b a n n n n = = ∑ ∑ ∞ = ∞ = 1 1 A and then: i. B A b a n n n ± = ± ∑ ∞ =1 ii. kA ka n n = ∑ ∞ =1 Theorem 2: The nth term test: ∑ ∞ n=1 n a is a series. If ∑ ∞ n=1 n a converges → 0 ⇒ an Equivalently: ∑ ∞ = ⇒ →/ 1 0 n n n a a diverges. (The converse of this theorem is not always true). Useful Remark: (UR): {an} is a sequence. 0 0 → ⇒ → n n a a . Equivalently, 0 0 →/ ⇒ →/ n n a a . Important Remark: (IR): ∑ ∞ n=1 n a is a series, N is a positive integer. Then, ∑ ∞ n=1 n a converges ⇔ ∑ ∞ n=N n a converges. Section 3: Definition 1: If n an ≥ ∀ 0 , we say ∑ ∞ n=1 n a is a series of nonnegative terms. If n an > ∀ 0 , we say ∑ ∞ n=1 n a is a series of positive terms. Corollary to IT: (CIT): ∑ ∞ n=1 n a is a series of nonnegative terms. Sn is the series of partial sum of ∑ ∞ n=1 n a . Then: ∑ ∞ n=1 n a converges ⇔ {Sn} is bounded from above. Theorem 1: Integral test: ∑ ∞ n=1 n a is a series of nonnegative terms. ( ) f x is a function that is continuous and decreasing on ( ,0 ∞) with ( ) n f n an ∀ = . Then: ∑ ∞ n=1 n a converges ⇔ ( ) ∫ ( ) ∫ ∞ → ∞ = b b f x dx x dx f 1 1 lim converges. Section 4: Theorem 1: Direct Comparison Test: (DCT): Given ∑ ∞ n=1 n a , ∑ ∞ n=1 nc , and ∑ ∞ n=1 n d series of nonnegative terms. Then: i. n c a n n ∀ ≤ : ∑ ∞ n=1 nc converges ⇒ ∑ ∞ n=1 n a converges. ii. n d a n n ∀ ≥ : ∑ ∞ n=1 n d diverges ⇒ ∑ ∞ n=1 n a diverges. Theorem 2: Limit Comparison Test: (LCT): ∑ ∞ n=1 n a series of nonnegative terms. ∑ ∞ n=1 nb series of positive terms. i. ∑ ∑ ∞ = ℵ = →∞ ⇒    < ∞ < = 1 1 and 0 lim n n n n n n n b a L L b a both converge of both diverge. ii. converges converges 0 lim 1 0 ∑ ∑ ℵ = ∞ = →∞ ⇒       = n n n n n n n a b b a iii. diverges diverges lim 1 1 ∑ ∑ ℵ = ∞ = →∞ ⇒       ∞ = n n n n n n n a b b a Section 5: Theorem 1: The Ratio Test: Suppose ∑ ∞ n=1 n a is a series of positive terms. Suppose n n n a a 1 lim + ρ = →∞ , then: i. ρ < 1⇒ series converges ii. ρ > 1⇒ series diverges iii. ρ = 1⇒ Test fails Theorem 2: The Root Test: Suppose ∑ ∞ n=1 n a is a series of nonnegative terms. Suppose n n n a ρ = →∞ lim , then: iv. ρ < 1⇒ series converges v. ρ > 1⇒ series diverges vi. ρ = 1⇒ Test fails Section 6: Definition 1: Suppose {an} is a sequence of positive terms, then: ( ) ... 1 4 3 2 1 1 + − + − = − ∑ ∞ = a a a a a n n n is called an alternating series. Theorem 1: The Alternating Series Test (AST): Suppose ( ) ∑ ∞ = + − ⇒    → ∀ ≤ ∀ > 1 1 1 0 0 n n n n n n n a a n a a n a converges. Theorem 2: Alternating Series Estimation Theorem (ASET): ( ) ∑ ∞ = + − 1 1 1 n n n a is an alternating series. {Sn} is the sequence of partial sums. Then: ( ) ∑ ∞ = + + = − ⇒    → ∀ ≤ ∀ > 1 1 1 0 0 n n n n n n n n error S a a n a a n a , with |error|≤ | first unused term| , and “error” has the same sign as the first unused term. Theorem 3: Absolute Convergence Test (ACT): ∑ ∞ n=1 n a is a series: ∑ ∑ ∞ = ∞ = ⇒ 1 1 converges converges n n n n a a . Definition 2: ∑ ∞ n=1 n a is a series that converges. If ∑ ∞ n=1 n a diverges and ∑ ∞ n=1 n a converges, we say that ∑ ∞ n=1 n a converges conditionally. Section 7: Definition 1: A series of the form ( ) ( ) ( ) ∑ ∞ = + − + − + = − 1 2 2 1 0 ... n n n a x c a c x c a x c is called a power series of center a, and of coefficient ,... , , 2 1 0 c c c Theorem 1: ( ) ∑ ∞ = − 1 n n n a x c is a power series, then: Theorem 2: ( ) ∑ ∞ = − 1 n n n a x c is a power series with radius of convergence R. Suppose ( R) R a a I + − = ; the internal of interval of convergence. Suppose ( ) ( ) ∑ ∞ = ∈ − = 1 with n n n I x a x c f x , then this function is infinitely differentiable on I, and ( ) ( ) I x a x nc x f n n n ∈ − = ∑ ∞ = − with ' 1 1 ( ) ( ) ( ) ∑ ∞ = − ∈ − − = 1 2 with 1 '' n n n I x a x c n n x f Theorem 3: ( ) ∑ ∞ = − 1 n n n a x c is a power series with radius of convergence R and suppose ( R) R a a I + − = ; the internal of interval of convergence, then: ( ) ∫ f x dx exists for all x in I, and ( ) ( ) ∑ ∫ ∞ = + − + + = 0 1 1 n n n a x n c c f x dx for x ∈ I Theorem 4: ∑ ∞ =0 n anxn and ∑ ∞ =0 n bnxn are 2 power series with radii of convergence 1 R and 2 R respectively. Suppose R is the minimum of 1 R and 2 R , then: ∑ ∑ ∑ ∞ = ∞ = ∞ =  =           0 0 0 n n n n n n n n n c x b x a x for x < R where ∑ = − = n k k n k n b a c 0 . Section 8: Definition 1: I is a open interval of center a. f(x) is a function which is infinitely differentiable on I. n a positive integer. Then: The Taylor polynomial of order n generated by f at x = a is: ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) n a n n a x n f a x x f a a x f f a P x − + + − + − + = ! ........ !2 '' ' 2 The Taylor series generated by ( ) f x at x = a is ( )( )( ) ( ) ( )( ) ( )( ) ∑ ∞ = + − + − + = − 0 2 ........ !2 '' ' ! n n n a x x f a a x f f a a x n a f Section 9: Taylor's Theorem: I is an open interval of center a. f(x) is a function which is infinitely differentiable on I. Suppose ( ) Pn x is a Taylor polynomial of order n generated by f at x = a . Then: ( ) ( ) ( ) x R P x x f n n + = with ( ) ( )( ) ( ) ( ) 1 1 ! 1 + + − + = n n n a x n c f x R for some c strictly between 0 and x. Section 10: Theorem 1: The Binomial Theorem: m ∈ IR , ( ) ∑ ∞ =       + = + + 2 1 1 n n m x n m m x for x < 1 where ( )( ) ( ) [ ] ! 1 2 ...... 1 n m m m m m n m − − − −  =      Section 11: Definition 1: I = [a, b] is a closed interval, and ( ) f x function defined on I. We say that f is piecewise continuous on I if: i. f has only finitely many points of discontinuity on . ii. If ( a b) x , 0 ∈ is a point of discontinuity of f, then ( ) ( ) f x x f x x − → − = 0 lim 0 and ( ) ( ) f x x f x x + → + = 0 lim 0 both exists. iii. ( ) ( ) f x a f a x + → + = lim and ( ) ( ) f x a f b x − → − = lim both exists. Definition 2: ( ) f x is a piecewise continuous function on [ 2,0 π] . The Fourier series of f is defined as ( ) ∑ ∞ = + + 1 0 sin cos 2 n n n nx b nx a a with: ( ) nx dx f x an ∫ π π = 2 0 cos 2 1 n = 1, 2, 3, …… ( ) nx dx f x bn ∫ π π = 2 0 sin 2 1 n = 1, 2, 3, …… Theorem 1: ( ) f x is a function defined on [ 2,0 π] if f and f' are piecewise continuous on [ 2,0 π] , then : continuity discontinuity ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           = π + π = π + + = + + + − − + + − ∞ =∑ 2 if 2 0 2 0 if 2 2 0 if is not a point of continuity of 2 if is a point of continuity of sin cos 2 1 0 x f f x f f f x f x x f f x x f nx b nx a a n n n