Determine whether the series ∑ n = 1 ∞ ( 2 3 + 1 3 ⋅ sin ⁡ ( n ) ) n n converges.

Question

Determine whether the series      ∑          n      =      1              ∞                  (              2        3            +              1        3            ⋅      sin      ⁡      (      n      )              )        n              n  converges.

Cahokiavv · Accepted Answer

Group the terms in the sum into &quot;tame&quot; and &quot;wild&quot; terms. The tame are defined as intergers that obey:            |        n  −      π    2    −  2  π  a            |        ≥      1          n              1                  /                4             integer) meaning they are &quot;far enough&quot; from making the sine equal to 1. Wilds are the non-tame integers.Using the following theorem about how close   π is to rational numbers: For every integers   p  ,  q such that       |    q      |    &amp;gt;  1            |        π  −      p    q              |        &amp;gt;      1                  |            q                        |                          20                      ,they show that the wild numbers       W    k   obey      W    k    ≥      1    2        k          77              /            76            W    k    ≥      1    2        k          77              /            76      meaning they are pretty scarce.By using simple small angle expansion of the sine function they show that the sum over the tame numbers is less than the sum of       e          −              n             and therefore converges.Because of their scarcity the sum over the wild numbers       W    k   is less than or equal to twice the sum over       1          k              77                  /                76             and therefore also converges. Thus the whole sum converges.

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