Chpter 6 Infinite series We briefly review this chpter in order to study series of functions in chpter 7. We cover from the beginning to Theorem 6.7 in the text excluding Theorem 6.6 nd Rbbe s test (Theorem 6.6) from section 6.2. 39
4 CHAPTER 6. INFINITE SERIES
Chpter 7 Functions Defined by Series nd Integrls We focus on sequences nd series of functions nd their uniform convergence. We only present summry for Section 7.3. The first two sections re rerrngements of those in the text. 7. Sequences of functions Let S be subset of R n nd for ech k N, f k is function from S to R m. Let f : S R m. We wnt to understnd wht f k converges to f s k mens nd the reltions between f k nd f. Pointwise convergence. We sy f k f pointwise on S if x S, lim f k (x) = f(x). However, this kind of convergence is not good enough when we consider the continuity, integrbility or differentibility of f. Uniform convergence. We sy {f k } converges uniformly on S to f if ε >, K >, k N : [ k > K = ( x S, f k (x) f(x) < ε) ]. Some books use the nottion: f k f. 4
42CHAPTER 7. FUNCTIONS DEFINED BY SERIES AND INTEGRALS We sy {f k } is uniformly convergent if its converges uniformly to some function f. Note tht if f k f uniformly then obviously f k f pointwise. Theorem 7.. The sequence {f k } converges to f uniformly on S if nd only if where lim M k =, (7.) M k = sup{ f k (x) f(x) : x S}. (7.2) Corollry 7.2. If there re C k such tht f k (x) f(x) C k, for ll x S, nd then f k f uniformly on S. lim C k =, In prctice, we find f(x) = lim f k (x) first (i.e. find pointwise limits of f k ), then prove or disprove (7.). As with sequences of vectors we hve the notion of Cuchy sequences. Definition 7.3. A sequence {f k } of functions on S is uniformly Cuchy if ε >, K >, k, j N : [ k, j > K = ( x S, f k (x) f j (x) < ε) ], or equivlently, ε >, K >, k, j N : [ k, j > K = sup{ f k (x) f j (x) : x S} < ε ]. Theorem 7.4. The sequence {f k } is uniformly convergent on S if nd only if it is uniformly Cuchy. Theorem 7.5. Suppose f k f uniformly on S. If ech f k is continuous on S, then so is f.
7.2. SERIES OF FUNCTIONS 43 Theorem 7.6. Let, b R nd < b. Suppose f k (for k N) nd f re integrble on [, b] nd f k f uniformly on [, b]. Then lim f k(x)dx = f(x)dx = lim f k (x)dx. Theorem 7.7. Let {f k } be sequence of functions of clss C on n intervl [, b]. Suppose tht f k f pointwise nd f k g uniformly on [, b]. Then f is of clss C on [, b] nd f = g. 7.2 Series of functions Let S R m (m is used to void the conflict with the following index n) nd f n : S R m. Define the prtil sum s k = f + f 2 +... + f k = k f n. (7.3) Then {s k } is sequence of functions nd is considered s n infinite series n= f n. We sy tht the series f n is uniformly convergent if the sequence of prtil sums {s k } is uniformly convergent. The limit of the series is, s usul, the limit of the prtil sums. Note: we cn consider (such s for power series) the infinite series of the form n= f n. Of course, in this cse, f n is defined for ll n. Theorem 7.8 (Cuchy criterion). The series f n is uniformly convergent on S if n only if ε >, K >, k, j N : j > k > K = sup{ f k+ (x) + f k+2 (x) +... + f j (x) : x S} < ε. (7.4) Corollry 7.9. If f n is uniformly convergent on S then ( ) sup{ f n (x) : x S} =, (7.5) lim n=
44CHAPTER 7. FUNCTIONS DEFINED BY SERIES AND INTEGRALS or equivlently, f n uniformly on S. Consequently, if f n does not converge to zero uniformly on S then f n is not uniformly convergent on S. Theorem 7. (The Weierstrss M-test). Suppose there re M n for n N such tht (i) f n (x) M n, for ll n N nd x S, nd (ii) M n <. Then f n is bsolutely nd uniformly convergent on S. Theorem 7.. Suppose ech f n is continuous on S nd f n converges to f uniformly on S. Then f is continuous on S. Theorem 7.2. Let, b R nd < b. Suppose ech f n is continuous on [, b] nd f n converges to f pointwise on [, b]. (i) If f n converges uniformly on [, b] then f k (x)dx = f(x)dx = f k (x)dx. (7.6) (ii) If ech f k is of clss C on [, b] nd f k is uniformly convergent then f C ([, b]) nd d dx [ ] f n (x) = f (x) = f k(x) (7.7) 7.3 Power series We consider the infinite series of the form n x n, n, x R. This is clled power series. Note tht when x = the series is convergent nd its limit is zero.
7.3. POWER SERIES 45 Theorem 7.3. For ny power series nx n, there is number R [, ] such tht we hve the following. (i) The series is bsolutely convergent for x < R. The series is divergent for x > R. The series is uniformly convergent on [ r, r] for ny r < R. (ii) Let f(x) = nx n whenever it is defined. Then The function f is continuous on ( R, R). The function f is of clss C on ( R, R), nd f (x) = n n x n, x ( R, R). (7.8) For, b ( R, R), the function f is integrble on [, b] nd f(x)dx = n x n dx. (7.9) In prticulr, for x ( R, R), we hve x n f(t)dt = n + xn+. (7.) Such R bove is clled the rdius of convergence of the series. Note tht R cn be which mens the series is divergent for ny x ; nd R cn be which mens the series is bsolutely convergent for ll x R. R cn be defined by R = sup{ x : n x n converges}. (7.) How to compute R? In fct R = /L where L [, ] is determined by L = lim n+ n whenever it exists, or in generl, L = lim sup or L = lim n n, (7.2) n n. (7.3)