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aceace

  • 3 years ago

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  1. aceace
    • 3 years ago
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    If \[ T_{n} = \frac{ n-1 }{ n }\], prove \[T_{n+1} - T _{n-1} = \frac{ 2 }{ n^2 -1 }\]

  2. Mikael
    • 3 years ago
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    This is nice exercise for the hands

  3. Mikael
    • 3 years ago
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    \[ T_{n+1} = \frac{n}{n+1}\]

  4. Mikael
    • 3 years ago
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    So \[ T_{n+1} - T_{n-1} = \frac{ n }{ n+1 } - \frac{n-1}{n} \]

  5. Mikael
    • 3 years ago
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    Sorry \[ T_{n+1} - T_{n-1} = \frac{ n }{ n+1 } - \frac{n-2}{n-1}\]

  6. Mikael
    • 3 years ago
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    Just bring to the common denominator by multiplying the denominators and recall that \[ (n+1)(n-1) = ...^2 - 1^2 \]

  7. aceace
    • 3 years ago
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    it seems like i cant go any further

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