4.12.E: Problems on Sequences and Series of Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
Complete the proof of Theorems 2 and 3.
Complete the proof of Theorem 4.
In Example (a), show that fn→+∞ (pointwise) on (1,+∞), but not uniformly so. Prove, however, that the limit is uniform on any interval [a,+∞),a>1. (Define "lim fn=+∞ (uniformly)" in a suitable manner.)
Using Theorem 1, discuss limn→∞fn on B and C( as in Example (a)) for each of the following.
(i) fn(x)=xn;B=E1;C=[a,b]⊂E1.
(ii) fn(x)=cosx+nxn;B=E1.
(iii) fn(x)=∑nk=1xk;B=(−1,1);C=[−a,a],|a|<1.
(iv) fn(x)=x1+nx;C=[0,+∞).
[Hint: Prove that Qn=sup1n(1−1nx+1)=1n.]
(v) fn(x)=cosnx;B=(0,π2),C=[14,π2).
(vi) fn(x)=sin2nx1+nx;B=E1.
(vii) fn(x)=11+xn;B=[0,1);C=[0,a],0<a<1.
Using Theorems 1 and 2, discuss limfn on the sets given below, with
fn(x) as indicated and 0<a<+∞. (Calculus rules for maxima and minima are assumed known in (v), (vi), and (vii).)
(i) nx1+nx;[a,+∞),(0,a).
(ii) nx1+n3x3;(a,+∞),(0,a).
(iii) n√cosx;(0,π2),[0,a],a<π2.
(iv) xn;(0,a),(0,+∞).
(v) xe−nx;[0,+∞);E1.
(vi) nxe−nx;[a,+∞),(0,+∞).
(vii) nxe−nx2;[a,+∞),(0,+∞).
[Hint: limfn cannot be uniform if the fn are continuous on a set, but limfn is not.
[For (v),fn has a maximum at x=1n; hence find Qn.]
Define fn:E1→E1 by
fn(x)={nx if 0≤x≤1n2−nx if 1n<x≤2n, and 0 otherwise
Show that all fn and limfn are continuous on each interval (−a,a), though limfn exists only pointwise. (Compare this with Theorem 3.)
The function f found in the proof of Theorem 3 is uniquely determined. Why?
⇒7. Prove that if each of the functions fn is constant on B, or if B is finite, then a pointwise limit of the fn on B is also a uniform limit; similarly for series.
⇒8. Prove that if fn→f( uniformly ) on B and if C⊆B, then fn→f (uniformly) on C as well.
⇒9. Show that if fn→f( uniformly ) on each of B1,B2,…,Bm, then fn→f (uniformly) on ⋃mk=1Bk.
Disprove it for infinite unions by an example. Do the same for series.
⇒10. Let fn→f( uniformly ) on B. Prove the equivalence of the following statements:
(i) Each fn, from a certain n onward, is bounded on B.
(ii) f is bounded on B.
(iii) The fn are ultimately uniformly bounded on B; that is, all function values fn(x),x∈B, from a certain n=n0 onward, are in one and the same globe Gq(K) in the range space.
For real, complex, and vector-valued functions, this means that
(∃K∈E1)(∀n≥n0)(∀x∈B)|fn(x)|<K.
⇒11. Prove for real, complex, or vector-valued functions fn,f,gn,g that if
fn→f and gn→g (uniformly) on B,
then also
fn±gn→f±g( uniformly ) on B.
⇒12. Prove that if the functions fn and gn are real or complex (or if the gn are vector valued and the fn are scalar valued), and if
fn→f and gn→g (uniformly) on B,
then
fngn→fg (uniformly) on B
provided that either f and g or the fn and gn are bounded on B (at least from some n onward); cf. Problem 11.
Disprove it for the case where only one of f and g is bounded.
[Hint: Let fn(x)=x and gn(x)=1/n (constant) on B=E1. Give some other examples.]
⇒13. Prove that if {fn} tends to f (pointwise or uniformly), so does each subsequence {fnk}.
⇒14. Let the functions fn and gn and the constants a and b be real or complex (or let a and b be scalars and fn and gn be vector valued ). Prove that if
f=∞∑n=1fn and g=∞∑n=1gn (pointwise or uniformly) ,
then
af+bg=∞∑n=1(afn+bgn) in the same sense.
(Infinite limits are excluded.)
In particular,
f±g=∞∑n=1(fn±gn) (rule of termwise addition)
and
af=∞∑n=1afn.
[Hint: Use Problems 11 and 12.]
⇒15. Let the range space of the functions fm and g be En(*or Cn), and let fm=(fm1,fm2,…,fmn),g=(g1,…,gn); see §3, part II. Prove that
fm→g (pointwise or uniformly)
iff each component fmk of fm converges (in the same sense) to the corresponding component gk of g; i.e.,
fmk→gk (pointwise or uniformly), k=1,2,…,n.
Similarly,
g=∞∑m=1fm
iff
(∀k≤n)gk=∞∑m=1fmk.
(See Chapter 3,§15, Theorem 2).
⇒16. From Problem 15 deduce for complex functions that fm→g (pointwise or uniformly) iff the real and imaginary parts of the fm converge to those of g (pointwise or uniformly). That is, (fm)re→gre and (fm)im→gim; similarly for series.
⇒17. Prove that the convergence or divergence (pointwise or uniformly) of a
sequence {fm}, or a series ∑fm, of functions is not affected by deleting or adding a finite number of terms.
Prove also that limm→∞fm (if any) remains the same, but ∑∞m=1fm is altered by the difference between the added and deleted terms.
⇒18. Show that the geometric series with ratio r,
∞∑n=0arn(a,r∈E1 or a,r∈C),
converges iff |r|<1, in which case
∞∑n=0arn=a1−r
(similarly if a is a vector and r is a scalar). Deduce that ∑(−1)n diverges. (See Chapter 3, §15, Problem 19.)
Theorem 4 shows that a convergent series does not change its sum if every several consecutive terms are replaced by their sum. Show by an example that the reverse process (splitting each term into several terms) may affect convergence.
[Hint: Consider ∑an with an=0. Split an=1−1 to obtain a divergent series: ∑(−1)n−1, with partial sums 1,0,1,0,1,…]
Find ∑∞n=11n(n+1).
[Hint: Verify: 1n(n+1)=1n−1n+1. Hence find sn, and let n→∞.]
The functions fn:A→(T,ρ′),A⊆(S,ρ) are said to be equicontinuous at p∈A iff
(∀ε>0)(∃δ>0)(∀n)(∀x∈A∩Gp(δ))ρ′(fn(x),fn(p))<ε.
Prove that if so, and if fn→f (pointwise) on A, then f is continuous at p.
[Hint: "Imitate" the proof of Theorem 2 .]