Find `F_("net") = GMm[ (1)/(r^(2))+ (1)/(2r^(2))+ (1)/(4r^(2))+ ...... "up to "oo]`.

sum _(r = 2) ^(oo) (1)/(r ^(2) - 1) is equal to :

find the sum of the series sum_(r=0)^(n)(-1)^(r)*^(n)C_(r)[(1)/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+(15^(r))/(2^(4r))... up to m terms ]

Prove that sum_(r=0)^(n)(-1)^(r)nC_(r)[(1)/(2^(r))+(3)/(2^(2r))+(7)/(2^(3r))+(15)/(2^(4r))+...up to mterms]=(2^(mn)-1)/(2^(mn)(2^(n)-1))

The sum sum_(r=2)^(oo)(1)/(r^(2)-1) is equal to

if r>1 and x=a+(a)/(r)+(a)/(r^(2))+.........oo,y=b-(b)/(r)+(b)/(r^(2))-(b)/(r^(2)) and z=c+(c)/(r^(2))+(c)/(r^(4))+............oo then (xy)/(z)

If a=sum_(n=r)^( oo)(1)/(r^(4)) then sum_(r=1)^(oo)(1)/((2r-1)^(4))=

sum_(r=1)^(oo)tan^(-1)((2)/(1+(2r+1)(2r-1)))

Find Lt(n rarr oo) sum_(r=0)^(n-1)(1)/(sqrt(n^(2) - r^(2))