Does ∑ n = 1 ∞ cos 2 ⁡ ( n + 1 ) n converge?

Question

Does       ∑          n      =      1              ∞                          cos                  2                    ⁡      (      n      +      1      )        n    converge?

humusen6p · Accepted Answer

∑          n      =      1              ∞                          cos                  2                    ⁡      (      n      +      1      )        n    clearly diverges. For any two consecutive integers, at least one of them is distant from a (non-integer) half-integer multiple of   π by at least       1    2  , which is more than       π    8  . That number of the pair will have a cosine greater than   cos  ⁡  (            3      π        8    ) in absolute value, and   cos  ⁡  (            3      π        8    )  =  sin  ⁡  (      π    8    )  &amp;gt;      1    2    sin  ⁡  (      π    4    )  =            2        4    &amp;gt;      1    3  . Thus, any two consecutive terms                     cos                  2                    ⁡      (      j      +      1      )        j    +                    cos                  2                    ⁡      (      (      j      +      1      )      +      1      )              j      +      1       will contribute at least       1          9      (      j      +      1      )       to the sum. Since   j  ≥  1, we have       1          j      +      1        ≥      1          4      j        +      1          4      (      j      +      1      )      , so       1          9      (      j      +      1      )        ≥      1          36      j        +      1          36      (      j      +      1      )       , i.e.                    cos                  2                    ⁡      (      j      +      1      )        j    +                    cos                  2                    ⁡      (      (      j      +      1      )      +      1      )              j      +      1        ≥      1          36      j        +      1          36      (      j      +      1      )      But this means      ∑          n      =      1              ∞                          cos                  2                    ⁡      (      n      +      1      )        n    ≥      1    36        ∑          n      =      1              ∞            1    n  which diverges.

Does <munderover> <mo movablelimits="false">∑ <mrow class="MJX-TeXAto

Answered question

Answer & Explanation

New Questions in Calculus 1