= 16 - 40 + 24
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- P(2) = (2)^4 - 5(2)^3 + 6(2)^2 + 4(2) - 8
- = 16 - 5 \times 8 + 6 \times 4 + 8 - 8
- = 16 - 40 + 24 + 8 - 8
- Question:** A mathematician is analyzing a topological space and needs to compute the sum of the series \( \sum_{k=1}^{50} \frac{1}{k(k+1)} \) as part of a homology calculation. Calculate this sum.
- Solution:** The given series is \( \sum_{k=1}^{50} \frac{1}{k(k+1)} \). We can simplify the terms using partial fraction decomposition:
- \frac{1}{k(k+1)} = \frac{A}{k} + \frac{B}{k+1}
