bài tập chuỗi số

nh ngha 3

nh ngha 3.Mt chui s c th vit di dng INCLUDEPICTURE "http://pdpstudio.org/phong/hoctoan/pages/giaitich1/chapV/ChuoiSo/chuoidandau_image002.gif" \* MERGEFORMATINET

trong cc scng du c gi l chui an du.nh l 13 (Du hiu Leibnitz).Cho chui an duNu dy sl dy n iu, hi t ti gii hn khng th chui an duhi t.Cho chui s c du bt k

Lp chui

Chui tr tuyt il chui s dng. Nhiu khi, thay cho kho st chui c du bt k, c th kho st chui s dng bng cch xt chui tr tuyt i.nh l 14.Nu chuihi t th chuicng hi t.S dng du hiu Leibnitz, d dng chng minh chui an duhi t. Lp chui tr tuyt i, c chui iu haV chui iu ha phn k, nn mnh o ca nh l 14 khng ng. Nh vy, vi hai chui ss c ba kh nng:1.Hai chuivcng hi t;2.Hai chuivcngcng phn k;3.Chuihi t v chuiphn k.nh ngha 4.Chuic gi l chui hi t tuyt i nu chuihi t. Mt chui s hi t nhng khng hi t tuyt i c gi l chui bn hi t.Nh vy, nu chui s hi t tuyt i th hi t, trng hp ny c th xt chui tr tuyt i; nu chui bn hi t th cn n my du hiu hi t quan trng nh: du hiu Dirichlet, du hiu Abel.nh l 15 (Du hiu Abel).Gi sl chui hi t cnl mt dy n iu v b chn. Khi chui

l chui hi t.nh l 16 (Du hiu Dirichlet).Gi sl chui s c dy tng ring b chn, ngha l tn ti s thcsao cho

cnl mt dy s khng m n iu gim hi t v khng, ngha lvKhi chui

l chui hi t.

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Cc v d

36. Xt s hi t ca chui s

Hng dn. Ta c

Theo du hiu so snh, chui shi t. Vy chui s cho hi t tuyt i.37. Xt s hi t tuyt i ca chui s

Hng dn. Xt tng

Ta c

Do

Vy chui sc dy tng ring b chn. Hn na,

l dy n iu gim hi t v khng. Theo du hiu Dirichlet, chui s cho hi t.Chng minh tng t,

l chui hi t. Chui cho khng hi t tuyt i. Tht vy, gi s

Khi , vi mita c

Do

T suy ra

V chui iu ha phn k cho nn y l mt mu thun. Mu thun chng t chui

phn k. Vy chui cho bn hi t.38. Xt s hi t ca chui s

Hng dn. Dy s

l dy n iu tng v b chn biv chui

hi t (v d 37). Theo du hiu Abel, chui s cho hi t. Bng cch so snh vi chui phn k

bn c c th chng minh chui s

l chui bn hi t.39. Xt s hi t tuyt i, bn hi t ca chui s

Hng dn. Xt ba trng hp(a) Trng hpTrng hp ny s hng tng qut khng dn ti khng, theo iu kin cn, chui s cho l chui phn k.(b) Trng hpChui cho l chui an du. Xt hai dy sv.Dyb chn trn biv t s hng th ba tr i dy s gim dn ti. Trong khi , dy

n iu gim v khng, vy nn

l dy n iu gim v khng. Theo du hiu Leibnitz, chui s cho hi t.Tuy nhin, trong trng hp ny n khng hi t tuyt i v

(c) Trng hpChui s cho hi t tuyt i v

40. Xt s hi t tuyt i, bn hi t ca chui s

Hng dn. Ta c

Ch rng, dy sl dy n iu gim v khng v cc s hng ca dy u thuc khongDo , dy s

l dy n iu gim v khng. Theo tiu chun Leibnitz, chui s cho l chui hi t. Tuy nhin, chui khng hi t tuyt i v

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Bi tp t gii

41. Dng du hiu Leibnitz; du hiu so snh xt s hi t ca cc chui s(a)(b)(c)(d)(e)(f)42. Dng du hiu Leibnitz, du hiu Abel, du hiu Dirichlet xt s hi t tuyt i v bn hi t ca cc chui s(a)(b)(c)(d)43. Dng du hiu Dirichlet chng minh chui s sau hi t

44. Xt s hi t tuyt i v bn hi t ca chui s

45 Chng minh rng chui shi t tuyt i khi v ch khi chui shi t tuyt i.46. Chng minh rng chui shi t th chui shi t tuyt i.47. Chng minh rng nu hai chui s c s hng tng qutvhi t tuyt i th cc chui s sau hi t tuyt i(a) (b)

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Hng dn hoc p s41. Dng du hiu Leibnitz; du hiu so snh.(a) Chui hi t.(b)trong VyChui (b) hi t v chui th nht hi ttheo du hiu Leibnitz; chui th hai hi t theo du hiu so snh.(c) Chui phn dl chui an du. D thyhi t theo du hiu Leibnitz, v vy chui (c) hi t.(e)tKhi Chui th nht hi t theo du hiu Leibnitz; chui th hai hi t theo du hiu so snh (v)(f) a v (b).42. Dng du hiu Leibnitz, du hiu Abel, du hiu Dirichlet xt s hi t tuyt i v bn hi t ca cc chui s(a) Khichui phn k. Khichuihi t theo du hiu Leibnitz cn dy

bt u t s hng th ba tr i l dy n iu v b chn bi 1. Do chui hi t theo du hiu Abel. Trng hp ny,vnn chui khng hi t tuyt i.Khichui hi t tuyt i.(b)Chui hi t vchui th nht hi t theo du hiu Leibnitz; chui th hai hi t tuyt i (v d 37).Chui khng hi t tuyt i vchuiphn k; chuihi t (v d 37 ).c) Chui hi t tuyt i theo du hiu Dirichlet (xem v d 37).(d) Chuic dy tng ring b chn bi 2:

dykhng m, n iu gim v 0, cho nn, theo du hiu Dirichlet, chuihi t. Vy chui cho hi t tuyt i.43. tKhi , chuic dy tng ring b chn bi 2; dydng, n iu gim v 0. Do chui cho hi t (Dirichlet)44.Chuihi t theo du hiu Abel (xem v d 38),;chuihi t tuyt i (du hiu so snh). Vy chui cho hi t.Vchuiphn k (v d 37), nn chui cho bn hi t.45. Chng minh rng chui shi t tuyt i khi v ch khi chui shi t tuyt i.

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