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From the identity \(\sin^2(\phi) = rac{ - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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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.
A + B = 0, \quad 2A = 1 \Rightarrow A = \frac{1}{2}, \quad B = -\frac{...
\frac{1}{n(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \rig...
\sum_{n=1}^{50} \frac{1}{n(n+2)} = \frac{1}{2} \sum_{n=1}^{50} \left(...
This is a telescoping sum. Write out the first few terms:

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VÃ©rifions :
\[ S_{13} = rac{13}{2}(6 + 24) = rac{13}{2} imes 30 = 195 \]
Recalculons :
\[ n = 14 \]
\[ S_{14} = rac{14}{2}(6 + 26) = 7 imes 32 = 224 \]
La racine correcte est \( n = 15 \) :
\[ S_{15} = rac{15}{2}(6 + 28) = rac{15}{2} imes 34 = 255 \]
VÃ©rifions proprement :
\( n = 15 \) est correct :
\[ S_{15} = 210 \]

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