MATH 201: LIMITS. a k+1 = 1.04a k

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1 MATH 0: LIMITS. Sequences Definition (Sequences). A sequence a n is a real-valued function () T R; n a n whose domain T is an inductive subset of the set N of natural numbers. Definition (Terms of a sequence). The real-number values a n of a sequence () are known as the terms of the sequence. Example 3 (Powers of ). The sequence n of powers of is,, = 4, 3 = 8,... In this case, the domain is the inductive subset T = N of N consisting of the natural numbers (starting at 0 to obtain = 0 ). Example 4 (The harmonic sequence). The harmonic sequence n is the sequence,, 3, 4,... of reciprocals of positive integers. In this case, the domain T is the set of positive integers. In Examples 3 and 4, the sequences were given directly by listing their terms as n and. Sometimes, sequences are defined recursively. n Definition 5 (Geometric sequences). Let a and r be real numbers. Then the geometric sequence a n with initial term a and ratio r is defined recursively by { a 0 = a and () a k+ = a k r for natural numbers k. Example 6 (A savings account). When Sam was born, the grandparents put $0, 000 into a savings account returning 4% interest per year. Thus the dollar amount of money a n in the account on Sam s n-th birthday is given by a 0 = 0000 a k+ =.04a k and

2 MATH 0: LIMITS for natural numbers k. This is a geometric sequence with initial term 0, 000 and ratio.04. On Sam s -st birthday, the account is worth $, 788. Theorem 7 (Terms of a geometric sequence). For each natural number n, the term a n of the geometric sequence () is given by a n = ar n. Proof. By induction on n, using the recursive definition.. Limits of sequences The sequence of powers of (Example 3) is,, 4, 8, 6, 3, 64, 8, 56, 5, 04, 048,... As n increases towards infinity, the terms of the sequence increase rapidly towards infinity. On the other hand, the terms,, 3, 4, 5, 6, 7,... of the harmonic sequence (Example 4) gradually decline towords 0 as n increases towards infinity. Definition 8 (Tending to infinity). A sequence a n is said to tend to infinity (as n tends to infinity), or to have infinity as its limit, if the following is true: (3) K R, N N. n N, a n K. Symbolically, we write lim a n = to state that the sequence a n tends to infinity (as n tends to infinity). Note that the third quantifier phrase n N, in (3) may be rewritten as n {m N m N}, using set-builder notation, or as n {N, N +, N +, N + 3,... }, using the roster method. The property (3) states that, however high the bar K is set, we can always go far enough along the list of terms of the sequence (specifically, as far as the term a N ), so that from that point on, all the terms a n of the sequence are above the bar (Figure ).

3 K MATH 0: LIMITS 3 a N a N+ a N+ a n N N + N +... n Proposition 9. Figure. Sequence a n tending to infinity. lim n =. Proof. Suppose that a real number K is given. We must show that there is a natural number N so that n K for all n N. Case : K < 0. If K is negative, we may choose N = 0 to guarantee n K for n N, since all the terms n are nonnegative. Case : K 0. In this case, round K up to the next integer N. (For example, if K = 3, then 3 = rounds up to N =.) Note N K. Then n N implies n N K, so that n K and n ( K) = K. Thus n K for all n N, as required. To show that a sequence a n does not tend to infinity, we have to prove the negation (4) K R. N N, n N. a n < K of the property (3). As before, the third quantifier phrase n N. in (3) may be rewritten as using set-builder notation, or as using the roster method. n {m N m N}. n {N, N +, N +, N + 3,... }.

4 4 MATH 0: LIMITS Proposition 0. The sequence n cos nπ does not tend to infinity. Proof. Consider K = 0. For a given natural number N, choose an odd integer n greater than N. Then n cos nπ = n < 0 = K. Warning! Although the symbolic equation lim a n = is used to express the fact that a sequence a n tends to infinity, we cannot use the symbolic equation (5) lim a n to express the fact that a sequence a n does not tend to infinity. (This is one case in which x y is not the negation of x = y!) If the inequality (5) is claimed to hold, then part of the claim is that the left hand side of (5) exists (as a limit in the general sense of Definition 8 below). However, saying that a certain sequence has a limit, which happens to differ from infinity, is not the same as saying that the sequence does not tend to infinity. For example, a sequence may well have no limit at all, finite or infinite. 3. Limits of geometric sequences Theorem (Binomial Theorem). For a real number x and natural number n, we have n ( ) n ( + x) n = x l l l=0 ( ) n = + nx + x + + x n. The Binomial Theorem is proved by induction on n. Corollary (Linear/Binomial Inequality). For a positive real number x and natural number n, we have + nx ( + x) n. Proof. If the real number x is positive, then ( ) n (6) 0 x + + x n since the powers of x and the binomial coefficients are positive. Note that the right-hand side of the inequality (6) is 0 for n <. Now consider the equality (7) + nx = + nx.

5 MATH 0: LIMITS 5 Adding the equality (7) to the inequality (6), we obtain ( ) n (8) + nx + nx + x + + x n. Finally, using the Binomial Theorem, we may rewrite the right-hand side of the inequality (8) to obtain + nx ( + x) n, which is the Linear/Binomial Inequality. Theorem 3 (Limit of a geometric sequence). Suppose that a n is a geometric sequence with positive initial term a and ratio r >. Then a n tends to infinity as n tends to infinity. Proof. Let r = + x. Note that the real number x is positive. By Theorem 7, we have a n = ar n = a( + x) n. Given a real number K, choose a natural number N with N K a ax. Thus anx K a and a( + Nx) = a + anx K. Now for n N, the Linear/Binomial Inequality gives as required. a n = a( + x) n a( + nx) a( + Nx) K, Theorem 3 shows that the powers of (Example 3) tend to infinity: Set the initial term a = and ratio r = in the theorem. 4. Finite limits. Definition 4 (Tending to zero). A sequence a n is said to tend to zero (as n tends to infinity), or to have zero as its limit, if the following is true: (9) ε > 0, N N. n N, a n < ε. Symbolically, we write lim a n = 0 to state that the sequence a n tends to zero (as n tends to infinity).

6 6 MATH 0: LIMITS ε a N a N+ a n N N + N +... n a N+ ε Figure. Sequence a n tending to zero. The use of the Greek letter ε (epsilon) for the positive real number in the first quantifier clause of (9) is traditional. Think of ε as a tolerance, like the tolerance required of machine parts that are supposed to fit together. Intuitively, we are claiming to be able to get the sequence terms a n as close to zero as anybody would require. If they ask us to make the terms stay within a distance ε of zero, we guarantee that this will happen for all a n once the index n is no less than our specified number N. Proposition 5. The harmonic sequence tends to zero. Proof. For a real number ε > 0, choose a natural number N with Then for n N, we have n N > ε. n N > ε or n > ε, which gives n < ε after inversion. Finally, note that n = n

7 since n > 0, so we obtain for n N, as required. MATH 0: LIMITS 7 n < ε To show that a sequence does not tend to zero, we must establish the negation (0) ε > 0. N N, n N. a n ε. of the condition (9). Intuitively, the condition (0) states that there is some tolerance ε such that infinitely many terms of the sequence miss zero by at least that tolerance. Proposition 6. The sequence a n = n n + does not tend to zero as n tends to infinity. Proof. Let ε =. Given a natural number N, choose n to be the maximum of N and 3. In particular, n 3. Now n 3 n n + 3 (n ) = n n + n (n + ) n n +. Since n n + 0 in this case, we have n n + = n n +, so n n + means n n + ε for the chosen n N, as required.

8 8 MATH 0: LIMITS Note that the sequence a n of Proposition 6 does not tend to zero, despite the fact that its term a is actually zero. The final result compares with Theorem 3. Theorem 7 (Limit of a geometric sequence). Suppose that a n is a geometric sequence with ratio r satisfying r <. Then a n tends to zero as n tends to infinity. Proof. Let a be the initial term a 0 of the sequence. By Theorem 7, we have a n = ar n for all natural numbers n. If the ratio r is zero, all the terms after a 0 are zero. Given ε > 0, the inequality a n < ε holds for all n, so the sequence also tends to zero in this case. Now suppose that r = 0. Since r <, we have so r >, r = + x for a certain positive real number x. Given ε > 0, choose a natural number N with N > a εx x. Then for n N, we have n > a εx x nx > a ε + nx > a ε. Using the Linear/Binomial Inequality, we obtain so that r = ( + n x)n + nx > a ε, a n = ar n = a r n < ε, as required to show that a n tends to zero.

9 MATH 0: LIMITS 9 Definition 8 (General limits). Let L be a real number. A sequence a n is said to tend to L or to have L as its limit if the sequence a n L tends to zero. Symbolically, we write to mean that lim a n = L lim (a n L) = 0. General limits are studied in advanced calculus and analysis.