I have the series ∑ n = 1 ∞ 1 − cos ⁡ ( x n ) n and I need to show that it converges pointwise for every x ∈ R

Question

I have the series       ∑          n      =      1        ∞                      1        −        cos        ⁡                  (                      x                          n                                )                            n             and I need to show that it converges pointwise for every   x  ∈      R

Stevinivm · Accepted Answer

A little bit of footwork with Calculus can establish that:  cos  ⁡  (  x  )  ≥  1  −      1    2        x    2  which of course implies  1  −  cos  ⁡  (  x  )  ≤      1    2        x    2    .Using this as an upper bound gives            1      −      cos      ⁡              (                  x                      n                          )                    n        ≤                    x        2            n              2              n              =            x      2              2              n                  3                      /                    2                      .Use this for the comparison test.

I have the series <munderover> <mo movablelimits="false">∑ <mrow class="

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