I am trying to show that if u 0 = u 1 = u 2 = 0, and if, when n &gt; 1 u 2 n − 1 = − 1 n , u 2 n = 1 n + 1 n + 1 n n

Question

I am trying to show that if       u          0        =      u          1        =      u          2        =  0, and if, when   n  &amp;gt;  1      u          2      n      −      1        =                    −        1                    n              ,      u          2      n        =            1              n              +            1      n        +            1              n                  n

klemmepk · Accepted Answer

Combine terms to get      ∏          n      =      2        N        (    1    −          1              n              )        (    1    +          1              n              +          1      n        +          1              n                  n                      )    =      ∏          n      =      2        N        (    1    −          1              n        2              )    =            N      +      1              2      N        →      1    2    .I should add that no,       lim          n      →      ∞            u    n    =  0   does not alone establish convergence. Consider      ∏          n      =      1        N        (    1    +          1      n        )    &amp;gt;      ∑          n      =      1        N        1    n    →  ∞    yet        lim          n      →      ∞            1    n    =  0.Some form of alternating test would work, but is computationally overdoing it in my opinion.

I am trying to show that if u <mrow class="MJX-TeXAtom-ORD"> 0 </mrow> </msu

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