If `x=sum_(n=0)^ooa^n , y=sum_(n=0)^oob^n , z=sum_(n=0)^ooc^n , w h e r e ra ,b ,a n dc` are in A.P. and `|a|<,|b|<1,a n d|c|<1,` then prove that `x ,ya n dz` are in H.P.

If x=sum_(n=0)^(oo) a^(n),y=sum_(n=0)^(oo)b^(n),z=sum_(n=0)^(oo)(ab)^(n) , where a,blt1 , then

If a,b,c are proper fractiion are in H.P. and x sum_(n=1)^oo a^n, y=sum_(n=1)^oo b^n, z= sum_(n=1)^oo c^n then x,y,z are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

If x=sum_(n=0)^oocos^(2n)theta,y=sum_(n=0)^oosin^(2n)varphi,z=sum_(n=0)^oocos^(2n)thetasin^(2n)varphi,w h e r e0 < theta,varphi < pi//2 prove that x z+y z-z=x ydot

If x=sum_(n=0)^oocos^(2n)theta,y=sum_(n=0)^oosin^(2n)varphi,z=sum_(n=0)^oocos^(2n)thetasin^(2n)varphi , where 0lttheta,varphi ltpi//2 prove that x z+y z-z=x ydot

"For "0ltthetalt(pi)/(2) , if x=sum_(n=0)^(oo)cos^(2n)theta,y=sum_(n=0)^(oo)sin^(2n)phi,z=sum_(n=0)^(oo)cos^(2n)thetasin^(2n)phi , then

If x = underset(n-0)overset(oo)sum a^(n), y= underset(n =0)overset(oo)sum b^(n), z = underset(n =0)overset(oo)sum C^(n) where a,b,c are in A.P. and |a| lt 1, |b| lt 1, |c| lt 1 , then x,y,z are in

sum_(n=0)^(oo) ((log_ex)^n)/(n!)

If S_n=sum_(r=0)^n 1/(nC_r) and t_n=sum_(r=0)^n r/(nC_r), then t_n/S_n=

Let x= sum_(n=0)^oo (-1)^n (tantheta)^(2n) and y = sum_(n=0)^oo (costheta)^(2n) qhere theta in (0,pi/4) , then

If a_n=sum_(r=0)^n1/(nC_r), then sum_(r=0)^n r/(nC_r) equals