1. Guía N°2<br />Luisa Fernanda Sánchez Gómez <br />Criterio de la integral<br />1∞12dx <br />= limb=> ∞dxx <br />=limb=> ∞ln x1b <br />= limb-∞LN b-LN1° <br />= limb-∞LN b=∞ <br />Diverge <br />n=1∞nn+1 1∞x(x+1) dx<br />U=x+1<br />Du=Dx<br />X=u-1<br />u-1 u du <br />= uu du-1u du <br />u-Ln u=x-Lnx+1b1 <br />limn-∞b-Ln(b+1)-1-Ln2=∞ <br />DIVERGE<br />n=1∞1n2 1x2 dx<br />1xb1 <br />limb=∞1x b1 <br />limb-∞1b-1=-1 <br />n=1∞1n2 <br /> DIVERGE PORQUE SE ESTA TRABAJANDO CON NUMEROS POSITIVOS.<br />EJERCICIOS PARA RESOLVER EN CASA<br />n=1∞1n2+1<br />1∞1(x2+1) dx=arctan(x) <br />arctan(x)b1 <br />limb-∞arctanb-π4 <br />-π4 <br />n=1∞1(n2+1) <br />DIVERGE <br />n=1∞n(n2+1) 1∞x(x2+1) <br />U=x2+1<br /> dudx=2x<br />du2x=dx <br />1∞xu du2x <br /> =1/2 1u ½ <br />Ln u = ½ LnX2+1b1<br /> = ½ Lnb2+1- Ln 2<br />n=1∞n(n2+1) <br />DIVERGE<br />n=1∞n2n3+1<br />1∞x2(x3+1) <br />U= x3+1<br />dudx=3x2 <br />du3x2=dx <br />xu2du3x2 <br />131u=13Lnu=13 Lnx3+1 <br />13 lim b-∞(Lnb3+1-Ln 2)=∞ <br />DIVERGE<br />n=1∞e-n<br />1∞e-x= -e-x b1 <br /> lim b-∞(-e-b+ e-1)= e-1 <br />= 0.36 <br />n=1∞e-n <br />CONVERGE A 0.36<br />n=1∞1nLnn<br />1∞1xLnx <br />DIVERGE<br />Criterio del cociente<br />n=1∞2n2n!<br />an= 2n2n! an+1= 2n2n+1!<br />limn=∞2n2n+1n!2n2n! = limn=∞1n+1<br />n=1∞2n2n! <br />CONVERGE <br />n=1∞22n!nn<br />an=22n!nn<br />an+1= 2n2n+1n! =(n+1)n(n+1)<br />n=1∞2n2n! <br />CONVERGE<br />n=1∞3nn2nn<br />an= 3nn2nn<br />an+1= 3n+1n2nnn+1= 3n3n2nnn = 3nnnn<br />limn=∞3nnnn5nn2nn = limn=∞3nnn<br />n=1∞3nn2nn <br />CONVERGE <br />n=1∞e-n2<br />an= e-n2an= e-n+12<br />limn=∞e-n2e-n+12 =limn=∞e-n2e-n2e2 = limn=∞1e2<br />n=1∞e-n2 <br />CONVERGE <br />n=1∞n2n!<br />an= n2n! <br />an+1= (n+1)2(n+1)n! = (n+1)n!<br />limn=1(n+1n!n2n!) = limn=1n+1n2<br />n=1∞n2n! <br />CONVERGE<br />n=1∞-1nn!nn<br />an=(-1)nn!nn <br />an+1= -1nn+1n!n+1n+1 = -1nn+1n!n+1nn+1 = -1nn!n+1n<br />limn=1-1nn!nn-1nn!-1nn!=limn=∞n+1nnn<br />n=1∞(-1)nn!nn <br />DIVERGE <br />n=1∞nnn!<br />an=nnn! <br />an+1= (n+1)n+1(n+1)n! = n+1n(n+1)(n+1)n! = (n+1)nn!<br />limn=∞((n+1)nn!nnn!) <br />n=1∞nnn! <br />DIVERGE <br />Criterio de p-series<br /> n=1∞1ne p= 6 converge 6>1<br /> n=1∞1nsen(45°) p= sen 45°=0.8 diverge 0.8<1<br />Criterio de la raiz<br />n=1∞1lnn+12<br />limn=∞1(ln⁡(n+1))2 <br />= limn=∞1nn+1-n = limn=∞1lnn+1<br />n=1∞1(lnn+1)2 <br />CONVERGE <br />n=1∞2nnn<br />limn=∞n2nnn = limn=∞1n<br />n=1∞2nnn <br />CONVERGE <br />n=1∞n2nn<br />limn=∞n2n-n <br /> = limn=∞nn2nn-n<br />= limn=∞nn2-n<br />n=1∞n2nn <br />CONVERGE <br /> n=1∞e-n<br />limn=∞ne-n = limn=∞1e1<br />n=1∞e-n <br />CONVERGE <br />Convergencia absoluta<br />n=1∞-1n1n<br />an= 1n <br />an+1=1n+1<br />1n=1n+1 <br />N+1 > n tiene convergencia absoluta<br />limn=∞1n Su límite es 0.<br />Por el criterio de la integral no hay convergencia absoluta.<br />n=1∞-1nn!nn<br />an= n!nn <br />an+1= n+1n!n+1n+1 = n!n+1<br />n!(n+1)<n!nn<br />No se cumple la primera condición entonces no hay convergencia absoluta.<br />n=1∞-1n+1nn<br />an= 1nn <br />an+1= 1n+1n+1<br />1n+1n+1<1nn<br />No se cumple la primera condición entonces no hay convergencia absoluta.<br /> <br />