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📅 May 16, 2026 👤 scraface

May 16, 2026

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f(u) = u^2 - 1.
Now substitute $ u = x^2 - 2 $ into $ f $:
f(x^2 - 2) = (x^2 - 2)^2 - 1 = x^4 - 4x^2 + 4 - 1 = x^4 - 4x^2 + 3.
\boxed{x^4 - 4x^2 + 3}
Question: Compute $ \sum_{n=1}^{50} \frac{1}{n(n+2)} $.
Solution: We begin by simplifying the general term using partial fract...
\frac{1}{n(n+2)} = \frac{A}{n} + \frac{B}{n+2}.
Multiplying both sides by $ n(n+2) $:
= A(n+2) + Bn.
= An + 2A + Bn = (A + B)n + 2A.

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