Question

pblicz granicę ciągu

lim (2n+1) /2 - ( 1+3+5+...+2n-1) / (n+1)

Answer 1

1 + 3 + 5 + ... + 2n - 1 = 0,5*[ 1 + 2n - 1]*n = 0,5*2n*n = n^2

zaten

(2n + 1)/2 - n^2 /( n + 1) = [ (2n+1)*(n +1) -2 n^2]/[ 2*( n + 1)] =

= [ 2 n^2 + 2n + n + 1 - 2 n^2]/( 2n + 2) =( 3n + 1)/( 2n + 2)

lim [ (3n + 1)/(2n + 2)] =

n --> +oo

= lim [ ( 3 + 1/n) /( 2 + 2/n)] = 3/2

n --> + oo

bo  1/n --> 0   i    2/n --> 0, gdy n--> + oo