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Math Help - Induction on a sequence

  1. #1
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    Induction on a sequence

    Q: Use the principle of induction to prove that 1/1^2 + 1/2^2 + ... + 1/n^2 <= 2 - 1/n for all n within the set of Natural Numbers.

    My Solution: Let S = {n within N : {1/n^2} <= 2 - 1/n}

    1 is within N, and 1/1^2 <= 2 - 1/1 = 1, so 1 is within S.
    Let k be within S.

    Proof: (k+1) is within S...

    Any hints?

    KK
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Q: Use the principle of induction to prove that 1/1^2 + 1/2^2 + ... + 1/n^2 <= 2 - 1/n for all n within the set of Natural Numbers.

    My Solution: Let S = {n within N : {1/n^2} <= 2 - 1/n}

    1 is within N, and 1/1^2 <= 2 - 1/1 = 1, so 1 is within S.
    Let k be within S.

    Proof: (k+1) is within S...

    Any hints?

    KK
    The infinite series,
    \sum_{k=1}^{\infty}=1+\frac{1}{2^2}+\frac{1}{3^2}+  ...
    Is called the Basel Problem.
    It was first solve (non-rigorusly) by Euler.

    Basically what this says is that the sequence of partial sums,
    1
    1+\frac{1}{2^2}
    1+\frac{1}{2^2}+\frac{1}{3^2}
    ....
    Has a least upper bound which is \frac{\pi^2}{6}.

    Thus, if H_n represents the n-th partial sum we have (since it is an upper bound) that,
    H_n\leq \frac{\pi^2}{6}
    Subtract 1 from both sides,
    \frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\leq \frac{\pi^2}{6}\approx 1.64<2-1/n
    Which is true for n\geq 3.

    Note, if you wish I can show you how to prove the Basel sum using a Fourier series?
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