FINAL REVIEW FOR MATH 500 SHUANGLIN SHAO. The it Define a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. This definition is useful is when evaluating the its; for instance, to show 2n 3n + = 2 3. One can also apply the it theorems to evaluate the it as follows. 2n 3n + = 2 3 + n = 2 3. We talk about the it theorem for its. If a n = A and b n = B, then a n ± b n = A ± B a nb n = AB a n = A, if B 0. b n B The proof of this theorem is by use of the definition of its. There is also another useful theorem for evaluating the its, the squeezing theorem : If a n b n c n and a n = c n = A, then b n = A. This theorem can be also proven by using the definition of its. A typical example is as follows. sin n 2 = 0 n Date: April 24, 202.
because 0 sin n 2 n n, 0 = n = 0. 2. The monotone convergence theorem and the sequence The monotone convergence theorem says that, if a sequence {a n } is either monotone increasing and bounded above, or monotone decreasing and bounded below, then {a n } converges. We give an example to show how to apply this theorem. Question. Determine whether the sequence {a n }: a 0 = and a n+ = 3 + 2an converges or not. If it converges, then find the it. Proof. We first show that it is monotone increasing. I have seen two ways: the first way is by using the mathematics induction. Since a 0 =, then a = 3 + 2a 0 = 5. So a 0 a. We assume that a n a n. Then 3 + 2an 3 + 2a n. Then a n a n+. The second way I talked about it in class. We look at a n+ a n = 3 + 2a n 3 + 2a n = 2(a n a n ) 3 + 2an + 3 + 2a n. The denominator is always positive. We see that the sign of a n+ a n is determined by the previous difference, a n a n. Inductively, we see that it is determined by a a 0, which is positive. So a n+ a n, for all n. We then show that it is bounded. We guess the bound is 3. We also prove it by mathematics induction. a 0 = 3. We assume that a n 3. Then a n+ = 3 + 2a n 3 + 2 3 = 3. 2
So a n 3 for all n. Then we apply the it theorems to get the it. A = 3 + 2A, i.e., A 2 2A 3 = 0. Then But A > 0, so A = 3 or A =. A = 3. 3. Series The convergence of a series k= a k is determined by its partial sum, n S n = a k = a + a 2 + a n. k= If {S n } converges, then a k converges and S n = a k. Otherwise, we say that the series diverges. The definition often provides a way to tell whether a series converges or diverges. There are several well-known examples of convergent or divergent examples: the following series, n=, p >, and np n=2 n(ln n) p, p >. They are convergent by comparing to the integrals, both of which are finite because p >. n= x dx and p On the other hand, the following series are divergent,, 0 < p, and np n(ln n) p, 0 < p. n=2 They are divergent by comparing to the integrals, both of which are infinity because 0 < p. The geometric series is convergent. r n for r <. n= 3 x dx and p x(ln x) p dx, x(ln x) p dx,
We can prove this by setting Then S n = n r k. k= ( r)s n = (r + r 2 + r n ) (r 2 + + r n + r n+ ) = r r n+. Since r <, S n = r rn+ r S n = r r n+ r = r r. We know that the harmonic series n= n is divergent. However if we add signs to each term ( )n n to turn it into n, then the series ( ) n n= n is convergent. In this regard, there is a general theorem: Suppose the sequence {a n } satisfying a a 2 a n 0 and n 0 a n = 0, then the series n= ( )n a n is convergent. A consequence that n= a n is convergent is that the tail terms go to zero as n goes to infinity, i.e., () a n = 0. This is a necessary not sufficient condition for a convergent series, for example, for n= n, n = 0 but the series n= n is divergent. However the equation () can be used as a criterion to tell whether a series converges or not: for n= a n, if a n 0, then n= a n is divergent. An example is, n= 2n is divergent because 2 n =. Another example n= ( + n ) is divergent because + n = 0. 4. series of functions In the previous section, we talk about several theorems about the series n= a n. By adding a variable x, the series of functions n= a nx n is also a kind of series with terms a n x n. Obviously it is convergent if x = 0. In general, there is a quantity associated to each series of functions, the radius of convergence ρ, such that, for x < ρ, the series is absolutely convergent; for x > ρ, it is divergent. However, for x = ρ, we need to study it case by case. We use the ratio test to find ρ. For example, for which values of x, does the series x n n converge? Proof. For this example, a n = n. 4
Then a n+ a n n = =, i.e., q =. (n + ) Hence the radius of convergence ρ = q =. That is to say, for x <, the series n= a nx n is absolutely convergent. For x >, the series n= a nx n is divergent. For x =, x = or x =. For x =, the series becomes n, which diverges. However for x =, the series becomes ( ) n n= n, which is convergent because it is a series of alternating terms. To conclude, the series is convergent on [, ), and divergent for x > and x =. 5. Limits of functions, continuity and uniform continuity Let f : D R, and a R. We say that x a f(x) = A, if for every sequence {x n } D satisfying x n a, f(x n) = A. Note that the point a may not be in the domain D. If there exists two sequences {x n } and {y n } both converging to a, and its n a f(x n ) and f(y n ) are not equal, then x a f(x) does not exist. This is a typical way to show that a it does not exist. An example, f(x) = x : R \ {0} R, in this case, f(x) does not exist. x x 0 The continuity of functions can be defined in a similar way. We say that f is continuous at x = a, if for every sequence {x n } D satisfying x n a, f(x n) = f(a). Note that a is in the domain D. If f is continuous at every point of D, then f is continuous on D. We have learned two theorems about continuous functions. One is the intermediate value theorem, and the other is the maximum principle. In the final, you are not required to know the proofs of these two theorems, but you need to know how to apply them to solve the problems. There is another way to phrase the continuity of functions, which we call the ɛ δ formulation of continuity of functions. We say that f is continuous 5
at x = a: for any ɛ > 0, there exists δ > 0 such that for x a < δ, Here δ may depend on ɛ and a. f(x) f(a) < ɛ. The second definition has the advantage of generalizing to uniform continuity of functions. We say that f is uniformly continuous on D: for any ɛ > 0, there exists δ > 0 such that for x y < δ, f(x) f(y) < ɛ. In general, the uniform continuity of functions implies the continuity of functions, but the converse is not true. For instance, f(x) = x 2 : R R is continuous on R but not uniformly continuous. However on a closed interval, say [a, b], they are equivalent. 6. differentiability Having seen continuity of functions, we talk about another property of functions, differentiability. We say that f is differentiable at a point a, if the following it exists, If the it exists, we denote h 0 f(a + h) f(a). h f f(a + h) f(a) (a) =. h 0 h The definition is important when we want to say whether a function is differentiable at a point; also it will gives the formula of the derivatives. For instance, f(x) = x n with n, is differentiable at every point x and f (x) = nx n. Another example is, for f(x) = sin x, f (x) = cos x. The most common example of non-differentiable functions is f(x) = x ; for this function, although it is continuous at x = 0, we know that it is not differentiable at x = 0. Related to differentiability, there are two important theorems, Rolle s theorem and the mean value theorem. The proof of Rolle s theorem makes use of the maximum principle we just mentioned. Roughly speaking, for continuous functions over [a, b], there exists c and d in [a, b] such that f(c) = max a x b f(x), f(d) = min f(x). a x b At these points, one can show that f = 0. The Rolle Theorem can be used to prove the mean value theorem. 6
7. integrability Let f be a bounded real function. We say that f is Riemann integrable on [a, b], if L(f) = U(f), where, L(f) = sup{l P (f)}, U(f) = inf {U P (f)}. P P Here P is a partition of [a, b]. We have shown that continuous functions or monotone functions over [a, b] are integrable. The Dirichlet function f is not Riemann integrable, where {, for irrational x; f(x) = 0, for rational x. Related to Riemann integrals, the fundamental theorem of calculus is important. Let f be a continuous function on [a, b]. Then is differentiable at x (a, b) and F (x) = F (x) = f(x), and b a x a f(t)dt f(t)dt = F (b) F (a). Note that F is called the anti-derivative of f. The last equation in the Fundament Theorem of Calculus tells a way to compute integrals b a f(t)dt: we just need to find what are the antiderivative of f. For example, π 0 sin xdx = ( cos π) + cos 0 = 2. This is because the antiderivative of sin x is cos x. Department of Mathematics, KU, Lawrence, KS 66045 E-mail address: slshao@math.ku.edu 7