Prove that,
`underset(n=1)overset(oo)sum cot^(-1)(2n^(2))=(pi)/(4)`

Prove that underset(r=1)overset(n)sum "tan"^(-1)(2r)/(r^(4)+r^(2)+2)="tan"^(-1)(n^(2)+n+1)-(pi)/(4)

The sum of the series underset(n+1)overset(oo)sumsin((n!pi)/(720)) is -

Find the value of underset(r = 0) overset(oo)sum tan^(-1) ((1)/(1 + r + r^(2)))

The value of underset(n=1)overset(13)sum(i^(n)+i^(n-1)),=i=sqrt(-1) is -

Statement -1 : The sum of the series (1+(1+2+3+4)+(4+5+9)+(9+12+16)+(361+380+400) "is"8000 Statement -2 : underset(k=1)overset(n)sum[k^(3)-(k-1)^(3)]=n^(3) for any natural number n .

Prove that underset(rles)(underset(r=0)overset(s)(sum)underset(s=1)overset(n)(sum))""^(n)C_(s) ""^(s)C_(r)=3^(n)-1 .

The limit of underset(n=1)overset(1000)sum(-1)^(n)x^(n)" as "x to oo

Find the value underset(n rarr oo)("lim") underset(k =2)overset(n)sum cos^(-1) ((1 + sqrt((k -1) k(k + 1) (k + 2)))/(k(k + 1)))

If x = underset(n = 0)overset(oo)Sigma cos^(2n)theta, " " y = underset(n =0)overset(oo)Sigma tan^(2n) varphi and z = underset(n = 0)overset(oo)Sigma sin^(2n) theta tan^(2n) varphi (0 lt theta, varphi lt (pi)/(4)) then show that xyz = xy + yz - z.

i^(2)=-1,"then"underset(n=0)overset(225)sumi^(n) is -