Lösungen / Solutions/I1S2Ue/Ex13

L = Lektion/ Leçon   -- Sprechstunde / heures de consultation
W = Work / Arbeit / Travail
A  =Anhang/  Annexe / Appendice

Ueb/Ex 1      W

a

Sum[
      4*(1/3)^(k - 1)  + 3*(1/4)^(k - 1), {k, 1, 
        Infinity}]

b

Sum[(1 + k + k^2 + k^3)/(1 + k + k^2 + k^3 + k^4 + k^5) , {k, 
        1, Infinity}]

Konv. (p-Reihen!) /  Conv . (Séries "p"!)

c

Sum[(-1)^k  1/k , {k, 1, Infinity}]

Konv. (Leibniz!) /  Conv . (Leibniz!)

Sum[(-1)^k  1/(2  k + 1) , {k, 1, Infinity}]

Konv. (Leibniz!) /  Conv . (Leibniz!)

d

Sum[(1/(3 (1 + k))), {k, 1, Infinity}]

Sum[(1/(3 (4 + k))), {k, 1, Infinity}]

Sum[
      (1/(3 (1 + k))) - (1/(3 (4 + k))), {k, 1, 
        Infinity}]

(1/(3 (1 + k))) - (1/(3 (4 + k))) // Together

Sum[(1/((1 + k) (4 + k))), {k, 1, Infinity}]

Apart[1/((k + 1) (k + 4))]

e

Sum[(4  x)^k, {k, 0, Infinity}]

Konv., Wurzelkriterium /  Conv., Cauchy  ==>  x <  1/4

f

Sum[2^k/(k!), {k, 0, Infinity}]

(2^(k + 1)/((k + 1)!))/(2^k/(k!))  // 
      Simplify

(((2 k!)/((1 + k)!)) == 2/(k + 1))

Limit[(2/(1 + k)), k -> Infinity]

Konv., Quotiontenkriterium /  Conv., d'Alembert