UNIVERSIDAD ABIERTA PARA ADULTOS (UAPA)
Nombre: Eri J. Almonte Belliard
Matricula: 17!"1#
Asignatura: An$li%i% &atem$ti'o II.
Facilitador: Ri'ard Bae
Fecha: **+1+*,1-
Santiao/ Re02li'a Domini'ana
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I) 3om0lete 'orre'tamente la% %i4iente% a'ti5idade% a) E%'ri2 E%'ri2e e 4n 'on'e 'on'e0to 0to de de %4'e%i %4'e%i6n. 6n.
Una Una su suce cesi sión ón es un unaa func funció iónn cu cuyo yo do domi mini nioo es el co conj njun unto to de los los números naturales tomado generalmente sin el cero y cuyo recorrido es un conjunto cualquiera de números reales. La imagen de cualquier número natural n! se denota "or F #n$% a n y se le llama t&rmino n'&simo de la sucesión. La sucesión es un conjunto orde ordena nado do de nú núme mero ross qu quee "u "ued edee e( e("r "res esar arse se "o "orr e( e(te tens nsió iónn o "o "or r com"rensión. 2) E%'ri2e E%'ri2e la dieren'ia dieren'ia entre entre la% %4'e%ione% %4'e%ione% 'on5er 'on5erente ente 8 di5erente.
Las sucesiones con)ergentes son las sucesiones que tienen l*mite finito.
L*mite % +
L*mite % , Las sucesiones di)ergentes son las sucesiones que no tienen l*mite finito. L*mite % -
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/ea 0 un una serie infinita dada1 dada1 y 2/n3 la sucesión de de sumas "arciales "arciales que n%, define esta serie infinita. 4ntonces si lim /n1 e(iste y es igual a /1 decimos que la serie serie n5 - es con)ergente con)ergente y que s es la la suma de la serie infinita infinita dada. dada. -/ea 0 un una serie serie infinita infinita dada1 y 2/n3 la sucesión sucesión de sumas "arciales "arciales que n%, define esta serie infinita. 4ntonces si lim /n1 no e(iste1 decimos que la serie serie n5 - es di)er di)ergen gente te y la serie serie no tiene tiene una suma. suma. d) E%'ri2e E%'ri2e la% rela'ione rela'ione% % 94e %e de2en tener tener en '4enta '4enta en lo% '$l'4lo% '$l'4lo% de l:mite%.
e$ f$ g$ h$ i$ j$
r 6+ % +6r % +1 r7 + -.r % -r % -'r % r6- % +
e) De%'ri2e la% %erie% ininita% de t;rmino% 0o%iti5o%.
/i todos los t&rminos de una serie infinita son "ositi)os1 la sucesión de sumas "arciales les es creciente. 8or lo tanto1 el siguiente teorem rema res resulta inmediatamente a "artir de teoremas anteriores. Teorema < -= Una serie infinita de t&rminos "ositi)os es con)ergente si y solo
si su sucesión de sumas "arciales tiene una cota su"erior. su"erior. ) E>0li'a la 'on%i%ten'ia de %erie de t;rmino% neati5o%. neati5o%.
) E%'ri2e 4n 'on'e0to de %4'e%i6n de 4n'ione%.
/i una sucesión tiene l*mite1 se dice que es una sucesión con)ergente1 y que la sucesión con)erge o tiende al l*mite. l*mi te. 4n caso contrario1 la sucesión
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) Deine %erie de 4n'ione%.
/i #fn$ es una sucesión acotada de funciones reales sobre un conjunto 9 y 0an es absolutamente con)ergente1 entonces 0anfn es uniformemente con)ergente en 9. es una serie de "otencias y an se llama el n'&simo coeficiente de la serie. i)?34$ndo %e oriina la %erie de @o4rier
Las series de Fourier reciben su nombre en honor a ean';a"tiste ose"h Fourier #,< Fourier #,<=> =>',> ',>?+$ ?+$11 que hi@o hi@o im"ort im"ortant antes es contrib contribuci ucion ones es al estudi estudioo de las series series trigonom&tr trigonom&tricas icas11 qu quee "re) "re)ia iame ment ntee ha hab* b*an an sido sido co cons nsid ider erad adas as "or Leonhard Leonhard 4uler 1 ean le ond dBAlembert y dBAlembert y Caniel ;ernoulli. ;ernoulli.a Fourier introdujo las series con el "ro"ósito de resol)er la ecuación de conducción dell ca de calo lor r een una lDm lDmina ina de metal "ublicando sus resulta ltados en ,>+< Mémoire sur la propagation de la chaleur dans les les corps solides #BMemoria
sobre la "ro"agación del calor en los cuer"os sólidosB$1 y
"ublicando su Théorie analytique de la chaleur #BEeor*a anal*tica del calorB$ en ,>. Gdeas "re)ias en descom"oner una función "eriódica en la suma de sim"les funciones de oscilación datan desde el siglo GGG a.H.1 cuando astrón astrónomo omoss ant antigu iguos os "ro"us "ro"usier ieron on un mod modelo elo em" em"*ric *ricoo de mo)imie mo)imiento nto "lanetario con base en e"iciclo. e"iciclo.
II) Determine lo% 0rimero% t;rmino% de la% %i4iente% %4'e%ione%=
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III) Determine T" 8 TC en la %4'e%i6n an !n*1
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I) Determine el 5to término de la sucesión F(n) = n 2 + 2n + 1 es:
F(5) = n2 +2n+1 F(5)= 52 + 2(5) + 1
F(5) = 25 + 10 +1
F(5)= 36
II) ¿u!l es el 5to término de la sucesión 3" 6" #$$$% 5to término de la sucesión es 15
III) &om'ree la letra corresondiente corresondiente a la resuesta correcta:
i) ¿u!l de estas sucesiones es eométrica%
a)1, 2, 3 ...
b)2, 4, 8, 16 ...
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b)2, "1, 2, "1, 2, "1....
c)6, 4, 2 ...
d)"3, "2, "1 ...
iii) #l $l%&o t'&ino de una (ogesin ait&'%ca de 6 t'&inos es 3* + la dieencia co&$n de sus t'&inos es 5. ¿Cuál es el (i&e (i&e t'&ino!
a)5
b)"5
c)3*
d)6
i-) #l (i&e t'&ino de una (ogesin ait&'%ca ait&'%ca de 6 t'&inos es 3 + el $l%&o t'&ino 23. ¿Cuál es la dieencia co&$n de los t'&inos!
a)4
b)6
c)23
d)3
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*i) l término eneral en la sucesión &n = 1" ," 2-" $$$ es:
a)1n
b)n2
c)n3
d)nn
-ii) l inte(ola 3 &edios ait&'%cos ente 1 + 32 se ob%ene
a)1, 37/6, 68/6, 99/6, 32
b)1, 37/6, 68/6, 32
c)1, 37/6, 32
d)32, 37/6, 68/6, 99/6, 32
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i) ¿Cuál es el 5to t'&ino de la (ogesin 5, 1*, 2*, ...!
a)8*
b) 9 *
c)4*
.) /l interolar , medios eométricos #6 3 se o'ene:
a)48, 24, 12, 6
b)96, 48, 24, 12, 6, 3
c)96, 48, 24, 3
d)4, 24, 12, 6, 3
d)2*
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"$ ',+. ',+.'= '=.' .'I I.. Jeom&trica q$ ,,.,. ,.,.' 'I. I. Aritm&tica r$ 'K.' 'K.',,?.' ?.',,I. Aritm&tica
VII) En la %i4iente %4'e%i6n. ?34$nto% '4adro% tiene la i4ra 94e %i4e
/e utili@a la formula n* 1 En total la i4ra %i4iente tendr$ * '4adro%
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u$ ,6? 1 , 1 ?I. Jeom&tric Jeom&tricaa "or se di)ide di)ide "or , )$ K1 ,+1 ,KI. ,KI.
Aritm& Aritm&tic ticaa "or se suma suma fijo K
O$ 1 ,1 ,KI.
Aritm&tica Aritm&tica "or se suma fijo ?
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IF) Determine el t;rmino eneral/ le8 o 'riterio de orma'i6n de la% %i4iente% %4'e%ione%=
a$ 1 >1 >1 ,1 ,=I b$ ,1 P1 P1 P?1 P?1 1 PK1 PK1 P=I P=I c$ ,1 1 1 ,=I
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