[1+4+10+19+.........+((3n^(2)-3n+2))/(2)=],[[" (a) "(n(n+1)(2n+1))/(6)," (b) "(n^(2)(n+1)^(2))/(4)," (c) "(n(n+1)(n+2))/(6)," (d) "(n(n^(2)+1))/(2)]]

1^(3)+2^(3)+3^(3)+.....+n^(3)=(n(n+1)^(2))/(4), n in N

1^(3)+2^(3)+3^(3)+...+n^(3)=n^(2)((n+1)^(2))/(4)

1.3+2.4+3.5+....+n(n+2)=(n(n+1)(2n+7))/(6)

Prove that log_(e)((n^(2))/(n^(2)-1))=(1)/(n^(2))+(1)/(2n^(4))+(1)/(3n^(6))+

1^(3)+2^(3)+3^(3)+………….+n^(3)=(n^(2)(n+1)^(2))/4 forall n in N.

Find underset(n to oo)lim x_(n) if (a) x_(n)=(3n^(2)+5n+4)/(2+n^(2) (b) x_(n)=(5n^(3)+2n^(2)-3n+7)/(4n^(3)-2n+11) (c) x_(n)=(4n^(2)-4n+3)/(2n^(3)+3n+4) (d) x_(n)=(1^(2)+2^(2)+.....+n^(2))/(5n^(3)+n+1)

lim_(n rarr oo)((n^(3)-2n^(2)+1)^((1)/(2))+(n^(4)+1)^((1)/(3 ))))/((n^(6)+6n^(5)+2)^((1)/(4))-(n^(7)+3n^(2)+1)^((1 )/(5)))

1.2.3+2.3.4++n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/(4)

1.2+2.3+3.4+.........+n(n+1)=(1)/(3)n(n+1)(n+3)

(1^(3)+2^(3)+...+n^(3))/(1+3+5+...+(2n-1))=((n+1)^( 2))/(4)