If `sum _( k =1) ^(oo) (k^(2))/(3 ^(k))=p/q,` where p and q are relatively prime positive integers. Find the value of `(p+q),`

Let lim _( x to oo) n ^((1)/(2 )(1+(1 )/(n))). (1 ^(1) . 2 ^(2) . 3 ^(3)....n ^(n ))^((1)/(n ^(2)))=e^((-p)/(q)) where p and q are relative prime positive integers. Find the value of |p+q|.

The value of sum_(n=1)^(oo)(1)/((3n-2)(3n+1)) is equal to (p)/(q), where p and q are relatively prime natural numbers.Then the value of (p^(2)+q^(2)) is equal to

Suppose that a parabola y=ax^(2)+bx+c where a>0 and (a+b+c) is an integer has vertex ((1)/(4),-(9)/(8)). If the minimum positive value of 'a' can be written as (p)/(q) where p and q are relatively prime positive integers, then find (p+q).

If the value of lim_(x->0)( ((3/ x)+1)/((3/x)-1))^(1/x) can be expressed in the form of e^(p/q), where p and q are relative prime then find the value of (p+q) .

If sin(2cos^(-1)((1)/(sqrt(5))))+cos(2tan^(-1)((1)/(3)))=(p)/(q) where p and q are relatively prime then digit at units place of (p-q)^(2k+1),k in N, can be

If 2009 = p^a x q^b where p and q are prime numbers, then the value of (p+q) is