Calculate the limit of the string (an) that has the general term an=(1/n)*(square root(1 +1/n)+square root(1 +2/n)+...+square root(1 +n/n))The number of terms of string is infinite.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.