IMPORTANT THEOREMS CHEAT SHEET


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1 IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop. We will see. Now remember, I m just dog nd theorems re confusing for dogs. So plese tell Bronson if I messed something up. Theorem 1 (Mthemticl Induction). Suppose for ech n N there is sttement P (n). If (Bse) P (1) is true; (Inductive) P (n) is true implies P (n + 1) is true, then P (n) is true for every n N. My thoughts: If I et the first tret, nd I know tht if I et tret I will get to et the next tret, then I will eventully be ft nd hppy. Theorem 2 (Well Ordering Principle). Let S N be nonempty subset. Then S contins smllest element. (This is exercise in your book) My thoughts: If I hve bunch of trets of vrious sizes nd I tke only some of them, then there must be smllest tret. I prefer big trets but sometimes Bronson mkes me compromise. Theorem 3 (Binomil Theorem). Suppose x, y R nd n N. Then n ( ) n (x + y) n = x n k y k. k k=0 My thoughts: I don t see ny trets in binomils. Theorem 4 (Completeness). The rel numbers re complete. Tht is, every bounded bove subset A R hs lest upper bound. 1
2 2 BY DOUGLAS DANE Theorem 5 (Archimeden Property). For every rel number x R, there exists nturl number n N such tht x < n. My thoughts: If you thought you bought me the biggest tret, I ssure you tht you didn t nd it breks my hert. Theorem 6 (Irrtionlity of 2). There doesn t exist rtionl number r such tht r 2 = 2. My thoughts: Don t you dre give me only 2 trets. I need t lest more thn tht. Theorem 7 (Density of Q). Suppose < b re rel numbers. Then there exists rtionl number r Q such tht < r < b. Theorem 8 (Density of Irrtionl Numbers). Suppose < b re rel numbers. Then there exists n irrtionl number c such tht < c < b. Theorem 9 (Tringle Inequlity). If, b R, then () + b + b ; (b) b b. Prt () is clled the tringle inequlity nd prt (b) is clled the reverse tringle inequlity. My thoughts: The best wy to lern the tringle inequlity is to look t tringle shped trets. Theorem 10 (Uniqueness of Limits). If n nd n b, then = b. Theorem 11 (Convergent sequences re bounded). Suppose n then n is bounded. Theorem 12 (Squeeze Theorem). If n, b n, c n re sequences such tht b n n c n. Suppose lso tht b n L nd c n L. Then n L. Note: Since convergence is til end behvior property, we only need the inequlity to be true for ll terms of the sequence beyond strting point. My thoughts: If I m wlking down hllwy, I m too big to turn round, so I hve to go wherever the hllwy tkes me. Hopefully, there re trets t the end. Theorem 13 (Algebric Limit Theorem). Suppose n nd b n b, λ R, k N. () λ n λ; (b) n + b n + b; (c) n b n b; (d) n /b n /b provided b, b n 0 for ll n; (e) k n k ; (f) 1/k n 1/k provided tht if k is even then we need n 0 for ll n. Theorem 14 (Order Limit Theorem). Assume n nd b n b. () If n 0 for ll n N, then 0. (b) If n b n for ll n N, then b. (c) If there exists c R for which c b n for ll n N, then c b. Similrly, if n c for ll n N, then c. As with the cse of the squeeze theorem, the inequlities only need to hold for ll n lrge enough.
3 IMPORTANT THEOREMS CHEAT SHEET 3 Theorem 15 (Monotone Convergence Theorem). Every bounded monotone sequence is convergent. Theorem 16 (Nested Intervl Property). If I 1 I 2 I 3 I n is sequence of closed nd bounded intervls, then n I n. Tht is, there is t lest one point x in ll of the intervls. Theorem 17 (BolznoWeierstrss). Any bounded sequence hs convergent subsequence. Theorem 18 (Completeness of R). A sequence is Cuchy if nd only if it is convergent. Theorem 19 (Sequentil Criterion for Continuity). A function f : D R is continuous t if nd only if for ll sequences n converging to, the sequence of outputs f( n ) converges to f(). Theorem 20 (Algebric Continuity Theorem). Let f nd g be functions with common domin D. If f, g re continuous t D, then: () λf is continuous t for ny λ R; (b) f + g is continuous t ; (c) fg is continuous t ; (d) f/g is continuous t if g() 0. Theorem 21 (Continuity of Composite Functions). Suppose Rnge(g) Domin(f) which is to sy f nd g re composble. If g is continuous t nd f is continuous t g(), then f g is continuous t. Theorem 22 (Extreme Vlue Theorem). If f is continuous function on closed nd bounded intervl I, then f is bounded on I nd f ttins both minimum nd mximum vlue, i.e. if m, M re the infimum nd supremum of Rnge(f), then there exists x 1, x 2 I such tht f(x 1 ) = m nd f(x 2 ) = M Theorem 23 (Intermedite Vlue Theorem). Let f be continuous function on the closed intervl [, b]. Suppose y is ny number between f() nd f(b). Then there exists c [, b] such tht f(c) = y. Theorem 24. If f is strictly monotone on I nd the rnge f(i) is n intervl, then f is continous. Theorem 25 (Monotone Continuity). Suppose f is strictly monotone on closed intervl, then f hs continuous inverse. Theorem 26. Let f : I R be continuous function closed nd bounded intervl. Then f is uniformly continuous. Theorem 27 (Preservtion of Cuchy Sequences under Uniform Continuity). Suppose f : D R is uniformly continuous. Then if (x n ) is Cuchy Sequence, then (f(x n )) is Cuchy sequence. Theorem 28 (Sequentil Criterion for NonUniform Continuity). Let f : D R be continuous function. Suppose there exists ε 0 0 nd sequences (x n ), (y n ) such tht the sequence of distnces converges to 0: x n y n 0
4 4 BY DOUGLAS DANE but Then f is not uniformly continuous. f(x n ) f(y n ) ε 0 for ll n N. Theorem 29 (Preservtion of Continuity under Uniform Convergence). Suppose (f n ) re sequence of continuous function on D. If (f n ) f uniformly, then f is continuous. Theorem 30 (Sequentil Criterion for Uniform Convergence). Suppose (f n ) is sequence of functions converging pointwise to functin f nd (b n ) is sequence converging to 0. If f n (x) f(x) b n for ll x nd for ll n. Then (f n ) f uniformly. Theorem 31. Let f(x) = g(x) on n open intervl I except mybe t I. Then lim f(x) = lim g(x) x x nd if one doesn t exist, then the other doesn t exist s well. Theorem 32 (Sequentil Criterion for Functionl Limits). Suppose f is defined on n intervl I nd suppose I or is one of the endpoints of Ī. Then lim f(x) = L x if nd only if for ll sequences n with n for ll n, we hve (f( n )) L. Where if is n endpoint, we men the corresponding one sided limit. Theorem 33 (Algebric Limit Theorem for Functionl Limits). Suppose f, g re defined on n intervl I nd is point in Ī. Suppose lim x f(x) = K, lim x g(x) = L, nd λ R. Then () lim (λf)(x) = λk; x (b) lim (f + g)(x) = K + L; x (c) lim (fg)(x) = KL; x (d) If L 0, then lim (f/g)(x) = K/L. x Agin, it is understood tht if is n endpoint of Ī, then the limit must be onesided. Theorem 34 (Differentibility implies Continuity). If f is differentible t, then f is continuous t. Theorem 35 (Algebric Differentition Theorem). Suppose λ R nd f, g re defined on n open intervl I nd I is such tht f (), g () exist. Then () (Sclr Multipliction Rule) (λf) () = λ f (); (b) (Addition Rule) (f + g) () = f () + g (); (c) (Product Rule) (fg) () = f ()g() + f()g (); (d) (Quotient Rule) If g() 0, then (f/g) () = f ()g() f()g () g() 2. Theorem 36 (Chin Rule). Suppose g is defined on n open intervl I nd is differentible t I. If f is defined on n open intervl contining g() nd f (g()) exists, then (f g) () = f (g()) g ().
5 IMPORTANT THEOREMS CHEAT SHEET 5 Theorem 37 (Derivtives of Inverse Functions). If f is strictly monotone on n open intervl I contining, f is differentible t. Then the inverse function g is differentible t b = f() nd g (b) = 1 f () = 1 f (g(b)). Theorem 38 (Interior Extremum Theorem). Suppose f : [, b] R is continuous. Then f chieves its extreme vlues t criticl points. Here, criticl points re either endpoints, points c (, b) such tht f (c) = 0, or points c (, b) such tht f (c) doesn t exist. Theorem 39 (Rolle s Theorem). Suppose f : [, b] R is continuous nd f is differentible on (, b). If f() = f(b), then there exists c (, b) such tht f (c) = 0. Theorem 40 (Men Vlue Theorem). Suppose f : [, b] R is continuous nd f is differentible on (, b). Then there exists c (, b) such tht f (c) = f(b) f(). b Theorem 41 (Drboux). Suppose f : (, b) R is differentible. Then f stisfies the intermedite vlue property. Tht is, for ll x < y (, b), if f (x) < α < f (y), then there exists x < c < y such tht f (c) = α. Theorem 42. Suppose f : (, b) R is such tht f (x) = 0 for ll x (, b). Then f is constnt function. Tht is, there exists λ R such tht f(x) = λ for ll x (, b). Theorem 43. If f, g : (, b) R re differentible nd stisfy f (x) = g (x) for ll x (, b), then there exists λ R such tht f(x) = g(x) + λ for ll x (, b). Theorem 44. If f : [, b] R is continuous nd f is differentible on (, b). Then f is strictly incresing if nd only if f (x) > 0 for ll x (, b). Similrly, f is strictly decresing if nd only if f (x) < 0 for ll x (, b). Theorem 45. If f is differentible on possibly infinite open intervl (, b) nd f (x) is bounded on (, b). Then f is uniformly continuous on (, b). Theorem 46 (Cuchy s Men Vlue Theorem). Suppose f, g : [, b] R re continuous nd differentible on (, b). Suppose g (x) 0 for ll x (, b). Then there exists c (, b) such tht f(b) f() g(b) g() = f (c) g (c). Theorem 47 (L Hôspitl s Rule). Let f, g be differentible function on possibly infinite open intervl (, b) nd suppose u = + or b. Suppose g(x) nd g (x) re nonzero on ll of (, b) nd either () lim x u f(x) = 0 = lim x u g(x), or (b) lim x u f(x) = ± = lim x u g(x). Then provided lim x u f (x) g (x) exists. f(x) lim x u g(x) = lim f (x) x u g (x)
6 6 BY DOUGLAS DANE Theorem 48 (εcriterion for Integrbility). The Riemnn integrl of f on [, b] exist if nd only if for ll ε > 0, there is prtition P of [, b] such tht U(f, P) L(f, P) < ε. Theorem 49 (Sequentil Criterion for Integrbility). The Riemnn Integrl for f on [, b] exists if nd only if there exists sequence of prtions P n of [, b] such tht lim(u(f, P n ) L(f, P n )) = 0. Theorem 50 (Monotone Integrbility). If f is monotone on closed nd bounded intervl [, b], then f is integrble on [, b]. Theorem 51 (Continuous Integrbility). If f is continuous on [, b], then f is integrble on [, b]. Theorem 52 (Linerity of Integrl). If f nd g re integrble on [, b] nd λ R. Then () λf is integrble on [, b] nd (b) f + g is integrble on [, b] nd λf = λ (f + g) = Theorem 53 (Order Preservtion of Integrl). Suppose f, g re integrble on [, b] nd f(x) g(x) for ll x [, b]. Then f(x)dx f. f + g(x)dx. Theorem 54 (Intervl Additivity). Suppose b c nd f is bounded on [, c]. Then: f = f = f + f + b If f is integrble on [, c], then it is lso integrble on [, b] nd [b, c] nd f = Theorem 55 (Men Vlue Theorem for Integrls). Suppose f is continuous on [, b]. Then there exists c [, b] such tht f(c) = 1 b f + b b f f. f. g. f(x)dx.
7 IMPORTANT THEOREMS CHEAT SHEET 7 Theorem 56 (Fundmentl Theorems of Clculus). (1) Let f be integrble on [b, c]. For [b, c] define function F : [b, c] R by F (x) = x f(t)dt. Then F is continuous on [b, c]. At ech point x (b, c) where f is continuous, F is differentible nd F (x) = f(x). (2) Let f be continuous on [, b] nd differentible on (, b) such tht f is integrble on [, b]. Then f (x)dx = f(b) f(). Theorem 57 (USubstitution). Let g : [, b] R be continuous on [, b] nd differentible on (, b). Assume tht F is differentible on g([, b]). If f(x) = F (x) for ll x g([, b]), then (f g) g = g(b) Theorem 58 (Integrtion by Prts). Suppose f, g : [, b] R re continuous nd differentible on (, b) such tht f, g re integrble on [, b]. Then fg nd f g re integrble on [, b] nd fg = fg b g() f. f g. Theorem 59 (Term Test). If series n converges, then lim n = 0. Theorem 60. If 0 R, then the geometric series r n converges to nd diverges otherwise. n=0 1 r if r < 1 Theorem 61. Suppose n is series nd b n is convergent series with b k 0 for ll n. Suppose lso tht there is lso constnt K such tht Then n converges bsolutely. n b n for ll n K. Theorem 62 (Integrl Test). Suppose f is positive nonincresing function on [1, ) nd k = f(k) for ech k N. Then the series k=1 k converges if nd only if the improper integrl f(x)dx converges. 1 Theorem 63 (Root Test). Given n infinite series k. Suppose ρ = lim k 1/k exists. Then the series converges bsolutely if ρ < 1 nd diverges if ρ > 1. If ρ = 1, then the series my converge or diverge. k=1 Theorem 64 (Rtio Test). Given n infinite series k, let r = lim k+1. k If the limit exists, then the series converges bsolutely for r < 1 nd diverges for r > 1. If r = 1, then the series my converge or diverge. k=1
8 8 BY DOUGLAS DANE Theorem 65 (Alternting Series Test). Let ( k ) be nonincresing sequence of nonnegtive numbers which converges to 0. Then the series ( 1) k k = k=0 converges. In fct, if s n is the nth prtil sum, then s s n n+1. Theorem 66 (Riemnn s Rerrngement Theorem). If k is conditionly convergent series, then for ech L R or L = ±, there exists rerrngement b j such tht bj = L. Theorem 67. If k is n bsolutely convergent series, then for ech rerrngement b j, we hve k = b j. Tht is, ny rerrngement of bsolutely convergent series converge to the sme number s the originl series. Theorem 68. If ech f k is continuous function on n intervl I nd the series f k (x) converges uniformly to g on I, then g is continuous on I. Theorem 69. If ech f k is continuous on [, b] nd f k converges uniformly to g on [, b], then g(x)dx = f k (x)dx. k=1 Theorem 70 (Weierstrss MTest). Suppose f k is sequence of functions defined on n interl I nd suppose f n (x) M n for ll x I nd n N for sequence of rel numbers M n. converges. Then f n (x) converges uniformly on I if M n Theorem 71 (Rdius of Convergence for Power Series). The rdius of convergence for power series c k (x ) k is 1 R = lim sup c k 1/k where R = (resp. 0) if lim sup c k 1/k = 0 (resp. ). Theorem 72 (Fundmentl Theorem for Power Series). Suppose the power series f(x) = ck (x ) k hs rdius of convergence R, i.e. converges uniformly for x ( R, + R), then f(x) converges uniformly on [ r, + r] for ll 0 < r < R; f(x) is continuous nd differentible on ( R, + R); x f(x)dx = c k k=0 k+1 (x )k+1 ; f (x) = k=1 kc k(x ) k 1. Theorem 73 (Tylor s Theorem). Let f be function with continuous derivtives up through order n + 1 in n open intervl I centered t. then for ech x I, f(x) = f() + f ()(x ) + + f (n) () (x ) n + R n (x), n!
9 IMPORTANT THEOREMS CHEAT SHEET 9 where R n (x) = f (n+1) (c) (x )n+1 (n + 1)! for some c I. The term R n (x) is clled the reminder. Theorem 74 (Lgrnge s Form for Reminder). If f is s in Tlor s Theorem, then the reminder R n (x) cn be written s R n (x) = 1 n! x (x t) n f (n+1) (t)dt.
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