Răspuns:
Explicație pas cu pas:
a) Este evidenta
b) Folosim a) :
1/(1*2) = 1/1 -1/2, 1/(2*3) = 1/2 -1/3, 1/(3*4) = 1/3 -1/4,....
S1 = 1 -1/2 +1/2 -1/3 +1/3 -1/4 +1/4 -...-1/2010 =
S1 = 1 -1/2010 = 2009/2010 < 1
c) Notam suma cu S3
1/(2*2) < 1/(1*2), 1/(3*3) < 1/(2*3), 1/(4*4) < 1/(3*4),
...,,1/(2009*2009)< 1/(2008*2009)
Adunam aceste inegalitati, obtinem:
S2 = 1/(2*2)+1/(3*3)+1/(4*4) +... +1/(2009*2009) <
< 1/(1*2)+1/(2*3)+1/(3*4)+...+1/(2008*2009) = S1 < 1
S3 = 1 + S2 < 1 + 1 = 2
S3 < 2
