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Thread: Test the convergence of a series

  1. #1
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    Test the convergence of a series

    The problem is short. Test the convergence of the following series (no need to find the value it converges to)
    $\displaystyle \sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$
    The task doesn't specify anything about k, so I am to assume that
    $\displaystyle k\in$\mathbb{R}$

    I've not been able to figure what to do with this one. I've briefly considered the ratio, root and integral method but didn't see how I could do anything with them so I tried to do it by definition
    $\displaystyle (\forall\epsilon>0)(\exists n_0\in\mathbb{N})(\forall n\geq n_0)(\exists p\in\mathbb{N})|S_{n+p}-S_n| < \epsilon$
    I've then tried to do sine sum to product but couldn't really see anything in there, and I've considered trying to add and subtract something inside the sin function and then use the angle sum identity but I can't think of anything I could add and subtract to achieve this.
    Does anyone have any suggestions or hints on how to solve this (or perhaps a solution, I won't argue against getting that)?

    Thank you.
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  2. #2
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    Re: Test the convergence of a series

    Quote Originally Posted by StereoBucket View Post
    The problem is short. Test the convergence of the following series (no need to find the value it converges to)
    $\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$
    The task doesn't specify anything about k, so I am to assume that
    $k\in \mathbb{R}$
    I've not been able to figure what to do with this one. I've briefly considered the ratio, root and integral method but didn't see how I could do anything with them so I tried to do it by definition
    $(\forall\epsilon>0)(\exists n_0\in\mathbb{N})(\forall n\geq n_0)(\exists p\in\mathbb{N})|S_{n+p}-S_n| < \epsilon$
    I've then tried to do sine sum to product but couldn't really see anything in there, and I've considered trying to add and subtract something inside the sin function and then use the angle sum identity but I can't think of anything I could add and subtract to achieve this.
    Does anyone have any suggestions or hints on how to solve this (or perhaps a solution, I won't argue against getting that)?

    Thank you.
    I have change "tex" and "/tex" to a dollar sign so that the new Latex generator will work.
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  3. #3
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    Re: Test the convergence of a series

    I would try the Alternating Series test. First, notice that as $n$ gets large, regardless of the value of $k$, $\sqrt{n^2+k^2} \sim n$.

    $\left|\sqrt{(n+1)^2+k^2} - \sqrt{n^2+k^2} \right|$ tends to 1 as $n \to \infty$.

    So, you can basically assume the sum is similar to $\displaystyle \sum_{n=1}^\infty \sin(n\pi)$. I think it will converge by the Alternating Series test.

    I think you can use uniform convergence of $\sin x$. So, given $\epsilon>0$, there exists $\delta>0$ such that $\left| \left| \sin (\pi x) \right| - \left| \sin \big(\pi (x+1+\delta) \big) \right| \right| < \epsilon$

    Then, use that $\delta$ to find $n_0 \in \mathbb{N}$ such that for all $n\ge n_0$, you have $\left| \sqrt{(n+1)^2+k^2}-\sqrt{n^2+k^2} - 1 \right| < \delta$
    Last edited by SlipEternal; Feb 27th 2018 at 01:33 PM.
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