A statistician develops a new robust estimator and models its bias under repeated samples by evaluating the sum \(\sum_{k=1}^{50} \frac{1}{k(k+2)}\). Compute this sum exactly.

["A New Robust Estimator and Its Bias: Unveiling the Exact Value of a Key Summation", "In the evolving landscape of statistical methodology, robust estimators play a crucial role in minimizing the influence of outliers and improving inference reliability. Recently, a pioneering statistician has introduced a novel robust estimator tailored for heavy-tailed distributions, enhancing resilience in small-sample settings. A foundational step in validating its performance involves analyzing the expected behavior of solvable mathematical expressions—such as the exact evaluation of the sum (\sum_{k=1}^{50} \frac{1}{k(k+2)}). This computation not only demonstrates precision in statistical modeling but also underpins the estimator’s asymptotic properties.", "### The Summation: A Foundation for Robust Modeling", "Consider the sum\n[\nS = \sum_{k=1}^{50} \frac{1}{k(k+2)}.\n]\nThis expression arises naturally when analyzing the influence function or expected bias of certain robust estimators, particularly those relying on weighted averages or inverse power trends. To compute (S) exactly, we begin with partial fraction decomposition.", "We rewrite the general term using identities:\n[\n\frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2}.\n]\nMultiplying both sides by (k(k+2)) gives:\n[\n1 = A(k+2) + Bk.\n]\nExpanding:\n[\n1 = Ak + 2A + Bk = (A + B)k + 2A.\n]\nMatching coefficients:\n- (A + B = 0),\n- (2A = 1) → (A = \frac{1}{2}), so (B = -\frac{1}{2}).", "Thus,\n[\n\frac{1}{k(k+2)} = \frac{1}{2} \left( \frac{1}{k} - \frac{1}{k+2} \right).\n]", "### Transforming the Sum", "Substitute into (S):\n[\nS = \sum_{k=1}^{50} \frac{1}{2} \left( \frac{1}{k} - \frac{1}{k+2} \right) = \frac{1}{2} \sum_{k=1}^{50} \left( \frac{1}{k} - \frac{1}{k+2} \right).\n]", "This is a telescoping series. Write out the first few and last few terms to observe cancellation:\n[\n\begin{aligned}\n&\left( \frac{1}{1} - \frac{1}{3} \right) + \left( \frac{1}{2} - \frac{1}{4} \right) + \left( \frac{1}{3} - \frac{1}{5} \right) + \left( \frac{1}{4} - \frac{1}{6} \right) + \cdots B\ \ldots\ + \left( \frac{1}{49} - \frac{1}{51} \right) + \left( \frac{1}{50} - \frac{1}{52} \right) \\n& = \frac{1}{1} + \frac{1}{2} - \frac{1}{51} - \frac{1}{52}.\n\end{aligned}\n]", "Higher terms cancel due to the telescoping structure: (-\frac{1}{3}) cancels with (+\frac{1}{3}), (-\frac{1}{4}) with (+\frac{1}{4}), and so on, leaving only the first two positive terms and the last two negative terms.", "Thus,\n[\n\sum_{k=1}^{50} \left( \frac{1}{k} - \frac{1}{k+2} \right) = 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52}.\n]", "Now compute:\n[\nS = \frac{1}{2} \left( \frac{3}{2} - \frac{1}{51} - \frac{1}{52} \right).\n]", "Compute the fractions:\nFirst, combine (\frac{1}{51} + \frac{1}{52}):\n[\n\frac{1}{51} + \frac{1}{52} = \frac{52 + 51}{51 \cdot 52} = \frac{103}{2652}.\n]", "Then:\n[\nS = \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right).\n]", "Convert (\frac{3}{2}) to year fraction with denominator 2652:\n[\n\frac{3}{2} = \frac{3 \cdot 1326}{2 \cdot 1326} = \frac{3978}{2652}.\n]", "Now:\n[\n\frac{3978}{2652} - \frac{103}{2652} = \frac{3875}{2652}.\n]", "Then:\n[\nS = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304}.\n]", "This fraction is already in simplest form (check gcd: 3875 and 5304 share no common factors ≥ 2; 3875 = 5³×31, 5304 not divisible by 5 or 31).", "Thus, the exact value of the sum is\n[\n\boxed{\frac{3875}{5304}}.\n]", "### Interpreting the Result in Statistical Context", "This precisely computed sum serves as a building block in deriving the expected bias and asymptotic efficiency of the new robust estimator. By analyzing how such algebraic constants behave under repeated sampling—via telescoping summation—statisticians validate the estimator’s numerical stability across data partitions. The method exemplifies how foundational mathematical manipulation supports theoretical advancement in robust statistics.", "In practice, such exact evaluations ensure transparency, reproducibility, and efficient computation—essential qualities when deploying modern robust tools. This example underscores that even seemingly algebraic exercises are deeply intertwined with real-world statistical innovation.", "Conclude: Mastery of exact summation is not merely academic—it is a critical lever in designing estimators that perform reliably under uncertainty, a hallmark of 21st-century statistical science."]