Evaluate the limit as n approaches infinity of sum from k=1 to n of n / (n2+k2).
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Step-by-Step Solution
Step 1: Rewrite the sum in a suitable form
To convert the sum into a form that can be recognized as a definite integral, we first divide the numerator and denominator by n2 This allows us to express the term inside the sum in terms of nk.
Step 2: Simplify the expression
After dividing by n2, we simplify the expression. The n in the numerator cancels with one of the n 's in the denominator, leaving n1 outside the term 1 + (nk)^2$.
Step 3: Convert the sum to a definite integral
We recognize this limit of a sum as a definite integral. Here, nk is replaced by x ,andn1isreplaced by dx. The limits of integration are from x=limn→∞n1=0 to x=limn→∞nn=1.
Step 4: Evaluate the integral
The integral of 1+x21 with respect to x is tan−1(x). We now need to evaluate this antiderivative at the upper and lower limits of integration.
Step 5: Apply the limits of integration
We apply the Fundamental Theorem of Calculus by substituting the upper limit (1) and the lower limit (0) into the antiderivative and subtracting the results.
Step 6: Calculate the final value
We know that tan−1(1) 4π (since tan(4π) = 1) and tan−1(0)=0(sincetan(0)=0). Subtracting these values gives the final result.