A SLOWLY DECREASING SEQUENCE AND ITS APPLICATION Sangadji *
ABSTRAK BARISAN YANG TURUN PERLAHAN DAN APLIKASINYA. Dalam makalah ini terlebih dulu akan didefinisikan barisan yang turun perlahan. Beberapa contoh dari barisan tersebut akan dijelaskan. Teorema tauber dari Schmidt dan Vijayaraghavan kemudian diperkenalkan. Sebelum membuktikan hasil utama dari makalah ini, dua lemma perlu dibuktikan. Hasil utama dari makalah ini adalah aplikasi dari barisan yang turun perlahan pada teorema tauber dari Schmidt –Vijayaraghavan.
ABSTRACT A SLOWLY DECREASING SEQUENCE AND ITS APPLICATION. In this paper a slowly decreasing sequence is first defined. Several examples of slowly decreasing sequences are then explained. The tauberian theorem of Schmidt and Vijayaraghavan is then introduced. Before proving the main result of the paper, two lemmas need to be proved. The main result of the paper is an application of a slowly decreasing sequence on the tauberian theorem of Schmidt - Vijayaraghavan.
INTRODUCTION
Definition A sequence (sn) of real numbers is said to be slowly decreasing if
lim inf (s m − s n ) ≥ 0 n→ ∞
whenever
m ≥ n and lim
n →∞
m−n n
= 0.
Using this definition we can observe the following examples concerning slowly decreasing sequences. * Pusat Pengembangan Teknologi Informatika dan Komputasi – BATAN
Examples (i) (ii) (iii)
It is obvious that every convergent sequence of real numbers is slowly decreasing. It is also obvious that every monotonically increasing sequence of real numbers is slowly decreasing. Let s n = (−1) n . Then, ( s n ) is not slowly decreasing. Clearly,
lim inf (s m − s n ) = −2 < 0 , where m = n + t (n) , m ≥ n , lim
n →∞
n →∞
(iv)
t (n) n
=0.
The sequence (λ n ) defined below is not slowly decreasing, where
λn = {
n=2 k
1
if
0
otherwise .
for some integer k ≥ 0 ,
Clearly, lim inf (s m − s n ) = −1 < 0 , for m = n + t (n) , m ≥ n and n →∞
lim
n→ ∞
t (n) n
= 0.
A TAUBERIAN THEOREM The following tauberian theorem is due to Schmidt [2] and Vijayaraghavan [4]. Shortly after Schmidt obtained the result Vijayaraghavan gave a different proof. Wiener obtained more general tauberian theorems, of which the one of SchmidtVijayaraghavan follows as a special case.This theorem is a special case of results obtained by Wiener (see Theorem XXI, [5]).
Theorem 1 Let (sn) be a slowly decreasing bounded sequence of real numbers. If ∞
sn n x =s n = 0 n!
lim e − x ∑ x →∞
then
lim sn = s .
n →∞
In this paper we do not have to prove the tauberian theorem. Its proof is very difficult and lengthy. We will apply a slowly decreasing sequence on the tauberian theorem of Schmidt – Vijayaraghavan.
TWO LEMMAS AND THEIR PROOFS Before proving the main result of the paper we need to prove these two following lemmas.
Lemma 1 Let (bn) be a sequence or real numbers where bn = n ∈ N0, N 0 =N ∪{0} . Then,
1 n −x n e x dx for every n! ∫0
bn +1 − bn ≥ 0 for every n ∈ N. Proof For every n ∈ N we have n +1 1 e − x x n +1dx ∫ 0 (n + 1)! n n +1 1 1 = e − x x n +1dx + e − x x n +1dx ∫ ∫ (n + 1)! 0 (n + 1)! n n +1 − 1 n n +1 − x 1 = + x de e − x x n +1dx ∫ ∫ 0 n (n + 1)! (n + 1)! n +1 n 1 n 1 = − e − x x n +1 (n + 1)! + ∫ e − x x n dx + e − x x n +1 dx ∫ 0 0 n n! (n + 1)! n +1 1 n 1 = − e − n n n +1 (n + 1)! + ∫ e − x x n dx + e − x x n +1 dx ∫ n! 0 (n + 1)! n n +1 1 = − e − n n n +1 (n + 1)! + bn + e − x x n+1 dx. ∫ n (n + 1)!
bn +1 =
Using the fact that the function y = e − x x n +1 achieves its maximum at x = n + 1 , we get
bn+1 − bn =
n +1 1 e − x x n +1dx − e −n n n +1 (n + 1)! ≥ 0 ∫ n (n + 1)!
for every n ∈ N, and the assertion follows. ¾ Lemma 2 Let ( λ n ) be a sequence of real numbers where
λn =
2 1 ∞ f (r )r 2 n+1 e − r / 2 dr , n ∈ N n ∫ 2 n! 0
and f is a real-valued and bounded function on R. Then, the sequence ( λ n ) is a slowly decreasing sequence.
Proof Since f is a bounded function on R, there exists a positive constant M such that f ( x) ≤ M on R. For every k in N0,
1 ∞ −r k 1 ∞ −r k 1 k −r k e r dr = e r dr − e r dr = 1 − bk . k! ∫k k! ∫0 k! ∫0 We have ∞ 1 f ((2 r )1 / 2 )e − r {r k +1 − ( k + 1)r k }dr ∫ 0 (k + 1)! k +1 ∞ M M ≥ e −r {r k +1 − (k + 1)r k }dr − e −r {r k +1 − (k + 1)r k }dr ∫ ∫ (k + 1)! 0 (k + 1)! k +1 1 k +1 1 ∞ = M bk +1 − ∫ e − r r k dr − M 1 − bk +1 − ∫ e − r r k dr 0 k + 1 k! k! 1 k +1 1 k +1 = M bk +1 − bk − ∫ e −r r k dr − M (1 − bk +1 ) − (1 − bk ) + ∫ e −r r k dr k! k k! k
λ k +1 − λ k =
= 2 M (bk +1 − bk ) −
2M k!
∫
k +1
k
e − r r k dr .
Using Lemma 1, we get
λ k +1 − λ k ≥ −2 Mck , where c k = e − k k k k! . Take n ∈ N0 , m = n + t (n) , m > n and t (n) show that lim inf (λ m − λ n ) ≥ 0 .
n → 0 as n → ∞ . We will
n→∞
Using the previous result and noting that (ck) is a monotonically decreasing sequence we get m −1
m −1
k =n
k =n
λ m − λ n = ∑ (λ k +1) − λ k ) ≥ −2 M ∑ c k ≥ −2 Mt (n)c n . Now we have
λ m − λ n ≥ −2Mt (n)c n = −2M
t (n) e − n n n 2πn . n! 2πn
Finally using Stirling’s formula and noting that t (n)
2πn → 0 as n → ∞ , we
obtain that lim inf (λ m − λ n ) ≥ 0 , i.e., the sequence (λ n ) is slowly decreasing. ¾ n→∞
MAIN RESULT Combining our previous results, the main result of the paper is the following theorem.
Theorem 2 Let (λ n ) be a sequence of real numbers where
λn =
2 1 ∞ f (r )r 2 n+1 e − r / 2 dr , n ∈ N0, n ∫ 2 n! 0
and f is a real-valued and bounded function on R. If
λn n x =λ, n = 0 n! ∞
lim e − x ∑ x →∞
then
lim λ n = λ . n →∞
Proof Suppose
λn n x =λ, n = 0 n! ∞
lim e − x ∑ x →∞
where
λn =
2 1 ∞ f (r )r 2 n+1 e − r / 2 dr , n ∈ N0. n ∫ 0 2 n!
By Lemma 2 this sequence (λ n ) of real numbers is slowly decreasing. Using the tauberian theorem of Schmidt – Vijayaraghavan (Theorem 1), we get
lim λ n = λ , n →∞
and the assertion follows. ¾
CONCLUSIONS (i)
Let (λ n ) be a sequence of real numbers. If λ n → 0 as n → ∞ . it is easy to show that
λn n x = 0. n = 0 n! ∞
lim e − x ∑ x →∞
(ii)
The converse of the above statement is false. Consider the sequence 1 if n = 2 k for some integer k ≥ 0 ,
λn = {
0
otherwise .
Then ∞ k λn n 1 x = e− x ∑ k x( 2 ) = 0 , n = 0 n! k = 0 ( 2 )! ∞
lim e − x ∑
x →∞
as was shown by Stroethoff in [3]. (iii)
Using the main result of the paper (Theorem 2), the converse mentioned above is true if (λ n ) is the sequence of real numbers where
λn =
1 ∞ 2 n +1 − r 2 / 2 f ( r ) r e dr 2 n n! ∫0
and f is a real-valued and bounded function on R.
REFERENCES
1. HARDY, G.H., Divergent Series, Clarendon Press, Oxford. (1949) 2. SCHMIDT, R., “Die Umkehrsätze des Borelschen Summierungsverfahrens”, Schriften der Königsberger Gelehrten Gesselschaft 1, (1925) p.202-256. 3. STROETHOFF, K., “The Berezin Transform and Operators in Spaces of Analytic Functions”, Linear Operators, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 38 (1997) p. 361-380. 4. VIJAYARAGHAVAN, T., “A Theorem concerning the Summability of Series by Borel’s Method”, Proc. London Math. Soc., (2) 27, (1928). p. 316-326. 5. WIENER, N., “Tauberian Theorems”. Ann. of Math., (2) 33, (1932), p. 1-100.
DISKUSI
MOCH SONHAJI 1. 2.
Kira-kira lim e − x x →∞
∞
λn
∑ n! x
n
= 0 , digunakan untuk apa?
n =0
Apakah ada Lemma yang dapat gunakan rumus tersebut?
SANGADJI 1. Dengan asumsi
lim λ n = 0 , maka selalu dapat disimpulkan bahwa n →∞
λn n x = 0, n = 0 n! ∞
lim e − x ∑ x →∞
λn n x n = 0 n!
Jadi
bila
kita
akan
∞
lim e − x ∑ x →∞
dan
dengan
diketahuinya
mencari
harga
lim λ n = 0 , n →∞
dari maka
λn n x =0 n = 0 n! ∞
lim e − x ∑ x →∞
2.
Bila proposisi di atas akan digunakan untuk pembuktian suatu teorema, maka proposisi tersebut dapat disebut lemma.
HASAN Dalam sistem kontrol modern selalu dibuat dalam model persamaan keadaan
x& = xA + Bu Biasanya kami menyelesaikan persamaan ini dengan deret Taylor kedua yang sebetulnya kurang teliti.
SANGADJI Makalah saya tidak ada hubungannya secara langsung dengan situasi tersebut.
DAFTAR RIWAYAT HIDUP
1. Nama
: SANGADJI
2. Tempat/Tanggal Lahir
: Solo, 16 Juni 1948
3. Instansi
: P2TIK - BATAN
4. Pekerjaan / Jabatan
: Ka.. Bidang Komputasi
5. Riwayat Pendidikan
: (setelah SMA sampai sekarang)
• FMIPA-UGM, Jurusan Matematika (1974)
(S1)
• Univ. of Arizona (USA), Jurusan Matematika (1988)
(S2)
• Univ. of Montana (USA), Jurusan Matematika (1997)
(S3)
6. Pengalaman Kerja
: 1974 – sekarang : Bekerja di BATAN
7. Organisasi Professional
:
• Himpunan Matematika Indonesia • Himpunan Fisika Indonesia
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