Compute \(\sum_{k=1}^{50} \frac{1}{k(k+1)}\).
["# Computing the Sum: (\sum_{k=1}^{50} \frac{1}{k(k+1)}) — A Powerful Mathematical Trick", "### Introduction\nCalculating sums involving fractions often reveals elegant patterns that simplify otherwise tedious calculations. One classic example is the sum\n[\n\sum_{k=1}^{50} \frac{1}{k(k+1)},\n]\nwhich demonstrates a powerful application of partial fractions and telescoping series. In this article, we’ll explore how this sum can be efficiently computed using a well-known mathematical identity, and why this approach is valuable in algebra, number theory, and competitive mathematics.", "---", "### The Power of Partial Fractions\nAt first glance, the term (\frac{1}{k(k+1)}) appears complex, but it simplifies beautifully through partial fraction decomposition. This technique expresses a rational function as a sum of simpler, more manageable fractions.", "Let’s decompose:\n[\n\frac{1}{k(k+1)} = \frac{A}{k} + \frac{B}{k+1}\n]\nTo find (A) and (B), multiply both sides by (k(k+1)):\n[\n1 = A(k+1) + Bk\n]\nExpanding:\n[\n1 = Ak + A + Bk = (A + B)k + A\n]\nMatching coefficients, we get:\n- Constant term: (A = 1)\n- Coefficient of (k): (A + B = 0 \Rightarrow B = -1)", "Thus,\n[\n\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}\n]", "---", "### Transforming the Sum into a Telescoping Series\nNow substitute the decomposition into the original sum:\n[\n\sum_{k=1}^{50} \frac{1}{k(k+1)} = \sum_{k=1}^{50} \left( \frac{1}{k} - \frac{1}{k+1} \right)\n]\nWriting out the first few and last few terms:\n[\n\left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \cdots + \left( \frac{1}{50} - \frac{1}{51} \right)\n]\nNotice that most terms elegantly cancel—this phenomenon is called telescoping.", "After cancellation, only two terms remain:\n[\n\frac{1}{1} - \frac{1}{51}\n]\nSo the sum simplifies to:\n[\n\sum_{k=1}^{50} \frac{1}{k(k+1)} = 1 - \frac{1}{51} = \frac{50}{51}\n]", "---", "### Why This Method Matters\nThis technique highlights how decomposition and pattern recognition can turn complicated infinite or finite sums into instantly recognizable differences. It’s widely used:", "- In algebra to simplify rational expressions and evaluate sums.\n- In number theory, for analyzing discrete structures and series.\n- In programming and algorithm design, where efficient summation and recursion are crucial.\n- In competitive exams, where mastering such tricks improves speed and accuracy.", "---", "### Final Answer\n[\n\sum_{k=1}^{50} \frac{1}{k(k+1)} = \frac{50}{51}\n]\nThis elegant result stems from a timeless mathematical strategy—partial fractions and telescoping—showcasing how simplicity lies beneath complexity.", "---", "### Tips for Practicing\n- Always try partial fraction decomposition before diving into computation.\n- Check cancellation patterns after decomposition to identify telescoping behavior.\n- Apply the same method to sums like (\sum_{k=1}^{n} \frac{1}{k(k+2)}) or generalized forms.\n- Explore extensions using computer algebra systems to verify your results.", "---", "Keywords: (\sum_{k=1}^{50} \frac{1}{k(k+1)}), partial fractions, telescoping series, summation, mathematical identity, decomposing rational functions, discrete mathematics."]
