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7 SEMICONTINUOUS FUNCTIONS 31

(b) Deduce from (a) that there are the following three alternatives : either the series
(a_ne~^^ns) does not converge for any value of s, or it converges for all values of s, or
else there exists a real number cr₀ such that the series converges for &s > a₀ and does
not converge when &s < cr₀ - In the first (resp. second) case we put cr₀ = +00 (resp.
— oo). In all cases, the (extended real) number cr₀ is called the abscissa of convergence
of the series. The sum of the series is an analytic function of s in the region %s > a₀ .

JI-K»

series with general term a_ne~*^nS — b_n is convergent (with s real and >0), then we
have |S₀,ii| ^ K>^A"^S where K is a constant, by writing a_n = b_ne*"^s. Then show that if
y = lim sup (log |S₀, n|)/A_n, the series (a_ne~*^nS) converges when s = y + S, where 8 is

n~>oo

real and >0, by arguing as in (a).)

(d) Let CTI be the abscissa of convergence of the Dirichlet series with general term
\a_n\e~*^nS. Then a± ^ o-₀ . If a₀ < + oo, show that

^ ,. 1°S^W

(TI — o"₀ ^ lim sup — r — .

n-*oo A_n

(Remark that, given e > 0, we have \a_n\ <; e^(<T°^+e)An for all sufficiently large n.) Consider
the case where A_n = n (cf. Section 9.1 , Problem 1). Show that, for the series with
general term

(—IV

: _ IL. g-siogiogn

Vn
we have a₀ — — oo and cr_t = +00.

10. Let/be a real-valued function defined on the interval [0, + oo [, satisfying the following
conditions:

(D /(J+0£/(*)+/(0;

(2) there exists a number M > 0 such that |/(r)| ^ M/ for all /.
Show that the limits

_r _R _r /(O

lim — , p = lim -¹-

t-*0 t t-t' + oo t

exist and are finite, and that a/ <i/(0 < fit for all t ;> 0. (To establish the existence
of a it is enough to show that, if we take a — lim sup /(/)// (Problem 8), then

I-+0

a ^f(t)/t for all t > 0. Take a decreasing sequence (t_n) tending to 0, such that
\imf(t_n)/t_n = a, and let k_n be the integral part of tjt_n . Show that

The proof of the existence of lim f(t)/t is similar; put ft = lim ii

t-* + °o r-* + oo

11. Let/be a continuous mapping of an open set U in a Banach space E into a Banach
space F. For each x e U, let

\\y~X\\

y-*,y** y~

We have 0 g D~/(x)



	7 SEMICONTINUOUS FUNCTIONS 31 (b) Deduce from (a) that there are the following three alternatives : either the series (a_ne~^^ns) does not converge for any value of s, or it converges for all values of s, or else there exists a real number cr₀ such that the series converges for &s > a₀ and does not converge when &s < cr₀ - In the first (resp. second) case we put cr₀ = +00 (resp. — oo). In all cases, the (extended real) number cr₀ is called the abscissa of convergence of the series. The sum of the series is an analytic function of s in the region %s > a₀ . (c) Show that if o-₀ ^0, then <T_O = lim sup (log \|S_0>n\|)/A_n. (Show first that if the JI-K» series with general term a_ne~^nS — b_n* is convergent (with s real and >0), then we have \|S₀,ii\| ^ K>^A"^S where K is a constant, by writing a_n = b_ne"^s.* Then show that if y = lim sup (log \|S₀, n\|)/A_n, the series (a_ne~^nS)* converges when s = y + S, where 8 is n~>oo real and >0, by arguing as in (a).) (d) Let CTI be the abscissa of convergence of the Dirichlet series with general term \a_n\e~^nS.* Then a± ^ o-₀ . If a₀ < + oo, show that ^ ,. 1°S^W (TI — o"₀ ^ lim sup — r — . *n-oo A_n** (Remark that, given e > 0, we have \a_n\ <; e^(<T°^+e)An for all sufficiently large n.) Consider the case where A_n = n (cf. Section 9.1 , Problem 1). Show that, for the series with general term (—IV : _ IL. g-siogiogn Vn we have a₀ — — oo and cr_t = +00. 10. Let/be a real-valued function defined on the interval [0, + oo [, satisfying the following conditions: (D /(J+0£/(*)+/(0; (2) there exists a number M > 0 such that \|/(r)\| ^ M/ for all /. Show that the limits

	_r _R _r /(O lim — , p = lim -¹- *t-0 t t-t' + oo t** exist and are finite, and that a/ <i/(0 < fit for all t ;> 0. (To establish the existence of a it is enough to show that, if we take a — lim sup /(/)// (Problem 8), then I-+0 a ^f(t)/t for all t > 0. Take a decreasing sequence (t_n) tending to 0, such that \imf(t_n)/t_n = a, and let k_n be the integral part of tjt_n . Show that

	The proof of the existence of lim f(t)/t is similar; put ft = lim ii t-* + °o r-* + oo 11. Let/be a continuous mapping of an open set U in a Banach space E into a Banach space F. For each x e U, let

	\\y~X\\

	y-,y* y~ We have 0 g D~/(x)