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Thread: Show that a case is true for certain conditions

  1. #1
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    Exclamation Show that a case is true for certain conditions

    Part (i): Show that 1/((n+1)^2) < (1/n) - (1/(n+1)) for ALL positive integers n.
    part (ii): Show that 1/(3^2) + 1/(4^2) + 1/(5^2) + .... + 1/(n^2) < 0.45 - 1/n for ALL n>= 3.
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    Re: Show that a case is true for certain conditions

    Hey azollner95.

    Can you show us what you have tried?
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    Re: Show that a case is true for certain conditions

    Quote Originally Posted by azollner95 View Post
    Part (i): Show that 1/((n+1)^2) < (1/n) - (1/(n+1)) for ALL positive integers n.
    part (ii): Show that 1/(3^2) + 1/(4^2) + 1/(5^2) + .... + 1/(n^2) < 0.45 - 1/n for ALL n>= 3.
    i) $\dfrac{1}{(n+1)^2}\le\dfrac{1}{n(n+1)}=\dfrac{1}{ n}-\dfrac{1}{n+1}$
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    Re: Show that a case is true for certain conditions

    I believe our professor wanted us to use induction (if that is even possible for these kind of problems). Can you explain in words Plato how you got that answer?

    Any hints on how to start part (ii)?
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    Re: Show that a case is true for certain conditions

    Quote Originally Posted by azollner95 View Post
    I believe our professor wanted us to use induction (if that is even possible for these kind of problems). Can you explain in words Plato how you got that answer?

    Any hints on how to start part (ii)?
    Yes. Induction is one plausible route for finding such proofs. But you certainly do not need induction to prove part i.

    As for part ii, why not start by seeing whether you can prove it for n = 3.
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