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sparkling water pros and cons - Coaching Toolbox

📅 May 4, 2026 👤 scraface

May 04, 2026

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Multiplying through by \(k(k+1)\) gives:
= A(k+1) + Bk = Ak + A + Bk
= (A + B)k + A
For the equation to hold for all \(k\), we set up the system:
A + B = 0 \quad \text{and} \quad A = 1
Solving this, we find \(A = 1\) and \(B = -1\). Thus:
Now, substitute this into the sum:
This is a telescoping series, where most terms cancel out:
After cancellation, we are left with:
Thus, the sum is:

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