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A science policy analyst is evaluating a proposal involving the sum of a series for technological forecasting. Compute \(\sum_{n=1}^{50} \frac{1}{n(n+1)}\).
The series \(\sum_{n=1}^{50} \frac{1}{n(n+1)}\) is a telescoping series. We simplify each term using partial fractions:
Partial fraction decomposition:
\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}
Write the series using partial fractions:
\sum_{n=1}^{50} \left( \frac{1}{n} - \frac{1}{n+1} \right)
Observe the telescoping nature:
The series expands as:
Simplify the result:
A science policy analyst is reviewing a proposal for a rectangular solar panel installation. The perimeter of the rectangle is 80 meters. What is the largest possible area of the rectangle?