Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Large numbers wikipedia , lookup
Law of large numbers wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Non-standard calculus wikipedia , lookup
Collatz conjecture wikipedia , lookup
Fundamental theorem of calculus wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Hyperreal number wikipedia , lookup
Limits of Sequences of Real Numbers Sequences of Real Numbers Limits through Rigorous Definitions The Squeeze Theorem Using the Squeeze Theorem Monotonous Sequences Index FAQ Sequences of Numbers Definition A sequence x1,x2 ,x3, is a rule that assigns, to each natural number n, the number xn. Examples Index 1 1 1 1 1, 2 , 4 , 8 , 2 1,1.4,1.41,1.414,1.4142, 3 1, 3,5, 7,9, FAQ Limits of Sequences Definition A finite number L is the limit of the sequence x1,x2 ,x3, if the numbers xn get arbitrarily close to the number L as the index n grows. If a sequence has a finite limit, then we say that the sequence is convergent or that it converges. Otherwise it diverges and is divergent. Examples Index 1 1 1 1 The sequence 1, , , , 2 4 8 0 and its limit is 0. converges FAQ 1 1 1 The sequence 1, , , , 2 4 8 0 and its limit is 0. Index converges FAQ Limits of Sequences 2 3 The sequence 1,1.4,1.41,1.414,1.4142, converges and its limit is 2. The sequence (1,-2,3,-4,…) diverges. Notation lim xn L n Index FAQ Computing Limits of Sequences (1) The limit of a sequence xn can be often computed by inserting n in the formula defining the general term xn . If this expression can be evaluated and the result is finite, then this finite value is the limit of the sequence. This usually requires a rewriting of the expression xn . Index FAQ Computing Limits of Sequences (1) Examples 1 2 1 1 1 1 The limit of the sequence 1, , , , n 1 is 0 because 2 4 8 2 1 inserting n to the formula xn n 1 one gets 0. 2 n2 1 The limit of the sequence 2 n 1 1 2 n2 1 n is 1 because rewriting 2 n 1 1 1 n2 1 and inserting n one gets 1. 1 n2 Index 0 FAQ Computing Limits of Sequences Examples continued 3 The limit of the sequence n 1 n n 1 n is 0 because of the rewriting n 1 n n 1 n n 1 n n 1 n 1 . n 1 n n 1 n Insert n to get the limit 0. Index FAQ Formal Definition of Limits of Sequences Definition A finite number L is the limit of the sequence x1,x2 ,x3, if 0 : n such that n n L xn . Example 1 0 since if 0 is given, then n n lim 1 1 1 0 if n n . n n Index FAQ Limit of Sums Theorem Assume that the limits lim xn x and lim y n y n are finite. Proof n Then lim xn y n x y . n Let 0 be given. We have to find a number n with the property n n xn y n x y . To that end observe that also Index 2 0. FAQ Limit of Sums Proof Hence there are numbers n1 and n2 such that n n1 xn x and n n2 y n y . 2 2 Let now n =max n1, n2 . We have n n xn y n x y xn x y n y 2 2 . By the Triangle Inequality Index FAQ Limits of Products The same argument as for sums can be used to prove the following result. Assume that the limits lim xn x and lim y n y Theorem n are finite. Then n lim xn y n x y . n Remark Observe that the limits lim xn y n and lim xn y n may exist n n and be finite even if the limits lim xn and lim y n do not exist. n n 1 . Then lim y n 0 and 2 n n Examples the limit lim xn does not exist. However, lim xn y n 0. Let xn 1 n and y n n n Index n FAQ Squeeze Theorem for Sequences Theorem Assume that n : xn y n zn and that lim xn lim zn a. n n Then the limit lim y n exists and n lim y n lim xn lim zn . n Proof n n Let 0. Since lim xn lim zn a, nx nz such n n that n nx xn a and n nz zn a . Let ny max nx , nz . Then n ny a y n max a xn , a zn . This follows since xn y n znn. Index FAQ Using the Squeeze Theorem Example Solution n! . n nn Compute lim This is difficult to compute using the standard methods because n! is defined only if n is a natural number. So the values of the sequence in question are not given by an elementary function to which we could apply tricks like L’Hospital’s Rule. n! Here each term k/n < 1. Observe that 0< n for all n 0. n Next observe that n ! 1 2 3 n 1 n 1 2 3 n n nnn nn n n n n 1 n 1 . n n n Hence 0 n! 1 . n n n 1 n! 0, also lim n 0 by the Squeeze Theorem. n n n n Since lim Index FAQ Using the Squeeze Theorem sin(n ) Does the sequence converge? n cos(n ) If it does, find its limit. Problem Solution 1 sin(n ) 1 We have Hence and 1 cos(n ) 1 for all n 2,3,4, . 1 sin(n ) 1 . n 1 n cos(n ) n 1 1 1 Since lim lim 0 we conclude that the sequence n n -1 n n -1 sin(n ) sin(n ) 0. converges and that nlim n cos(n ) n cos(n ) Index FAQ Monotonous Sequences A sequence (a1,a2,a3,…) is increasing if an ≤ an+1 for all n. The sequence (a1,a2,a3,…) is decreasing if an+1 ≤ an for all n. Definition The sequence (a1,a2,a3,…) is monotonous if it is either increasing or decreasing. The sequence (a1,a2,a3,…) is bounded if there are numbers M and m such that m ≤ an ≤ M for all n. Theorem A bounded monotonous sequence always has a finite limit. Observe that it suffices to show that the theorem for increasing sequences (an) since if (an) is decreasing, then consider the increasing sequence (-an). Index FAQ Monotonous Sequences A bounded monotonous sequence always has a finite limit. Theorem Let (a1,a2,a3,…) be an increasing bounded sequence. Proof Then the set {a1,a2,a3,…} is bounded from the above. By the fact that the set of real numbers is complete, s=sup {a1,a2,a3,…} is finite. Claim lim an s. n Index FAQ Monotonous Sequences A bounded monotonous sequence always has a finite limit. Theorem Let (a1,a2,a3,…) be an increasing bounded sequence. Proof Let s=sup {a1,a2,a3,…}. Claim lim an s. n Proof of the Claim Let 0. We have to find a number n with the property that n n an s . Since s sup an , there is an element an such that s an s. Since an is increasing n n s an an s. Hence n n an s . This means that lim an s. n Index FAQ