Estimating the Value of a Series — Question 2

Question 2 – Solution

Step 1: State the two-sided estimate.

For a positive decreasing function $f$ with $a_n=f(n)$, $$\int_{N+1}^{\infty}f(x)\,dx\le R_N\le\int_N^{\infty}f(x)\,dx.$$ Here $f(x)=x^{-2}$ is positive, continuous, and decreasing.

Step 2: Evaluate both integrals.

$$\int_{N+1}^{\infty}\frac{dx}{x^2}=\frac1{N+1},\qquad \int_N^{\infty}\frac{dx}{x^2}=\frac1N.$$ Thus $$\boxed{\frac1{N+1}\le R_N\le\frac1N}.$$

Step 3: Convert the tail bracket to a sum bracket.

Since $S=S_N+R_N$, $$\boxed{S_N+\frac1{N+1}\le S\le S_N+\frac1N}.$$ The bracket width is $1/[N(N+1)]$.