We'll re-write the general term of the string:
an = (1/n)*Sum sqrt(1 + k/n)
We'll identify a function f(k/n) = sqrt(1 + k/n)
an = (1/n)*Sum f(k/n)
If the function is continuous and it is, then the limit of the string is represented by the definite integral of f(x), where the limits of integration are x = 0 and x = 1.
lim an = Int f(x)dx
f(x) = sqrt(1+x) (the fraction k/n was substituted by x)
We'll put 1 + x = t
dx = dt
Int sqrt(1+x) dx = Int sqrt t dt = (2/3)*t^(3/2)
We'll apply Leibniz Newton to determine the value of the definite integral:
F(1) = (2/3)*2^(3/2)
F(0) = (2/3)*1^(3/2)
Int sqrt(1+x) dx = (2/3)*2^(3/2) - 2/3
Int sqrt(1+x) dx = (2/3)[2sqrt 2 - 1]
The limit of the string (an), represented by the definite integral of the function sqrt(x+1), is lim an = 2*[2sqrt 2 - 1]/3.