Determine whether ∑ k = 1 ∞ ( 1 − cos ⁡ ( 1 k ) ) converges or not

Question

Determine whether       ∑          k      =      1              ∞        (  1  −  cos  ⁡  (      1          k        )  ) converges or not

Hailee Henderson · Accepted Answer

1  −  cos  ⁡      (                  1                              k                                )    =  2      sin    2    ⁡      (                  1                              k                                )  Now using Limit comparison test with       1    k   we get      lim          k      →      ∞                  2              sin                  2                    ⁡              (                  1                      k                          )                    1      k        =  2      ∑          k      =      1              ∞            1    k   diverges . Hence By limit comparison test . The original series also diverges.

patzeriap0 · Accepted Answer

The serie is definitely non-negative. We can use the asynpotic criteria:  1  −  cos  ⁡      (          1              k              )    ∼      1    k  Because we know the followinf notable limit:      lim          t      →      0                  1      −      cos      ⁡      (      t      )              t      2        =  1Now, we can apply the comparison test with       ∑          n      =      1              +      ∞            1    n   that diverges. Thus, your serie diverges.

Determine whether <munderover> ∑ <mrow class="MJX-TeXAtom-ORD"> k

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