Problemas de para el Final Alvaro M. Naupay Gusukuma 12 de enero de 2017
´Indice general
´Indice general
Prefacio
iii
1. Series
1
2. Soluciones
1
Bibliograf´ ıa
3
i
ii
´Indice general
Prefacio
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iii
iv
Prefacio
1 Series
1. Determine el valor de la suma
∞
ln
k =1
(k + 1) 2 k (k + 2)
A) ln2
B)
D ) ln5
E)
ln3 1
C)
ln4
Soluci´on. ∞
2. Si
n=1
1 =
n2
S
∈
A)
S
4 4S D) 3
determine el valor de
∞
convergencia de nos de
R,
S
n=1
B) E)
1 , en t´ermi(2n − 1)2 S
2
C)
3S 4
S
8 Soluci´on.
1
2
Cap´ıtulo 1. Series
2 Soluciones
1. Note que ln
(k + 1) 2 (k + 1) = ln k (k + 2) k = ln
k+1 k
×
(k + 1) = ln (k + 2) k+2
+ ln
−
k+1 k
1
= ln
k+1
+ ln
k+1 k
f (k )
k+1 k+2 −
ln
k+2 k+1
f (k +1)
luego podemos aplicar la propiedad telesc´opica para series infinitas ∞
ln (k + 1) 2 = ln 1 + 1 k (k + 2) 1 k =1
Rpt.- ln 2
∞
2. Tenemos que
S
=
n=1 ∞
1 n2
= 1+
−
ım l´ ln
k →∞
k+2 k+1
= l n 2 − ln 1 = ln 2 Problema.
1 1 1 1 1 1 1 1 1 + + + + + + + + 22 32 42 52 62 72 82 92 102
...
∞
1 1 S = , donde P es la suma de los t´erminos de orden n)2 n2 (2 4 4 n=1 n=1 par la serie S . Adem´as note que la serie I de los t´erminos de orden impar est´a dado por 1 I = , que es justo lo que nos piden. n − 1)2 (2 n=1 Observe que
P
=
1
=
∞
Luego tenemos que Rpt.-
S
= P + I , es decir
I
= S −P =S
S −
4
3S 4
1
=
3S 4
. Problema.
2
Cap´ıtulo 2. Soluciones
Bibliograf´ıa
[1] Adomian, G. and Adomian, G.E. , A Global Method for Solution of Complex Systems , Mathematical Modelling, Vol5, 1984, pp. 251-63. [2] Adomian, G. Rach, R. and Sarafyan, D. , On the Solution of Equations Containing Radicals by the Decomposition Method, Journal of Mathematical Analysis and Applications, Vol. III No. 2, 1985, pp. 423-26. [3] Adomian, G. and Rach, R. , Polynomials Non-linearities in Differential Equations , Journal of Mathematics Analysis and Applications, Vol. 109 No. 1, 1985. [4] Adomian, G. , Non-linear Stochastic Dynamica Systems in Physical Problems , Journal of Mathematical Analysis and Applications, vol. III No. I, 1985. [5] George Adomian , Solving Frontier Problems of Physics: The Decomposition Method , Kluwer Academic Publisher, 1994. [6] Yves Cherruault , Convergence of Adomian’s Method , Medimat, Universit´ e de Paris, 1988. [7] Abdul-Majid Wazwaz , Partial Differential Equations and Solitary Waves Theory , Springer, 2009.
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