I wonder is it true that ( ∑ n : a n ≤ ϵ a n 2 ) ⋅ ϵ ( − 3 / 2 ) stays bounded as ϵ → 0for all a n that satisfies the above conditions.

Question

I wonder is it true that      (          ∑              n        :                  a          n                ≤        ϵ                    a      n      2        )    ⋅      ϵ          (      −      3              /            2      )        stays bounded as   ϵ  →  0for all       a    n   that satisfies the above conditions.

cuprins60 · Accepted Answer

All we need to know is that   0  ≤      a    n    ≤      n          −      2      . Let&#039;s let   0  &amp;lt;  ϵ  &amp;lt;  1, since we&#039;re taking   ϵ  →  0 anyways, and choose   N  =      ⌈          ϵ              −        1                  /                2              ⌉  . Moreover, let&#039;s let       n    0   be the first n so that       a                  n        0              &amp;lt;  ϵThen we have                              (                      ∑                                          a                n                            &amp;lt;              ϵ                                            a            n            2                    )                          ϵ                      −            3                          /                        2                                              =                  (                      ∑                          n              =                              n                0                                      ∞                                a            n            2                    )                          ϵ                      −            3                          /                        2                                                            =                  (                      ∑                          n              =                              n                0                                                    N              +              1                                            a            n            2                    +                      ∑                          n              =              N              +              1                        ∞                                a            n            2                    )                          ϵ                      −            3                          /                        2                                                            ≤                  (          (          N          +          1          −                      n            0                    )                      ϵ            2                    +                      ∑                          n              =              N              +              1                        ∞                                n                          −              4                                )                          ϵ                      −            3                          /                        2                                                                      ≤                      (            1            )                                    (          2          N                      ϵ            2                    +                      ∫                          N                        ∞                                x                          −              4                                           d          x          )                          ϵ                      −            3                          /                        2                                                            ≤                  (          2          N                      ϵ            2                    +                      N                          −              3                                )                          ϵ                      −            3                          /                        2                                                                      ≤                      (            2            )                                    (          2          (          1          +                      ϵ                          −              1                              /                            2                                )                      ϵ            2                    +          (                      ϵ                          −              1                              /                            2                                            )                          −              3                                )                          ϵ                      −            3                          /                        2                                                            ≤                  (          2                      ϵ            2                    +          2                      ϵ                          3                              /                            2                                +                      ϵ                          3                              /                            2                                )                          ϵ                      −            3                          /                        2                                                            =        3        +        2                  ϵ                                                  ≤        5            In step (1) we&#039;ve used the fact that       ∑          N      +      1        ∞    f  (  n  )  ≤      ∫    N    ∞    f  (  x  )     d  x and the (crummy) bound   N  +  1  ≤  2  N. In step (2) we&#039;ve replaced an N in the numerator by   1  +      ϵ          −      1              /            2       in the denominator by       ϵ          −      1              /            2      , since       ϵ          −      1              /            2        &amp;lt;  N  &amp;lt;  1  +      ϵ          −      1              /            2      . Lastly, since we&#039;re only considering   ϵ  &amp;lt;  1, we can get a uniform bound.

I wonder is it true that ( <munder> ∑ <mrow class="MJX-TeX

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New Questions in Calculus 2