Show that ∑ k = 1 ∞ sin k ⁡ ( k π / 6 ) diverges?

Question

Show that       ∑          k      =      1              ∞            sin    k    ⁡  (  k  π      /    6  ) diverges?

Trey Ross · Accepted Answer

Let       a    k    =      sin    k    ⁡  (  k  π      /    6  )As you noticed,       a          12      k        =  1 so       lim          k      →      ∞            a    k    ≠  0, which implies that the series       ∑          k      =      1        ∞        a    k   is divergent (by the Divergence Test).Now, let       b    k    =  sin  ⁡  (  k  π      /    7      )    k   and       c    k    =  sin  ⁡  (  k  π      /    7  ). The sequence       c    k   is 14-periodic, so   c  =  M  a  x  {      |        c    k        |    ,  k  ≥  1  } exists. Clearly,   c  &amp;lt;  1. We have       c    k    ≤  c  &amp;lt;  1 so      |        b    k        |    ≤      c    k  Since       |    c      |    &amp;lt;  1 is convergent, therefore       ∑          k      =      1        ∞        b    k   is absolutely convergent, hence convergent.

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