Does ∑ k = 1 ∞ 2 k − 1 x 2 k − 1 − 1 1 + x 2 k − 1 converges uniformly.

Question

Does       ∑          k      =      1              ∞                          2                  k          −          1                            x                              2                          k              −              1                                −          1                            1      +              x                              2                          k              −              1                                           converges uniformly.

minnegodks · Accepted Answer

The series does not converge uniformly on   (  0  ,  1  ) and hence also not on   (  −  1  ,  1  )To see this, let       f    n    (  x  )  =                    2                  n          −          1                    ⋅              x                              2                          n              −              1                                −          1                            1      +              x                              2                          n              −              1                                          . Then each       f    n   is continuous and (hence) bounded on   [  0  ,  1  ]Now, let   f  (  x  )  :=      ∑    n        f    n    (  x  ). If       ∑          n      =      1        N        f    n    →  f would hold with uniform convergence on   (  0  ,  1  ), it would follow that f is bounded on   (  0  ,  1  )But since each term of the sum is nonnegative for   x  ∈  (  0  ,  1  ), we have  ‖  f      ‖    sup    ≥  f  (  x  )  ≥      f    n    (  x  )  ≥                    2                  n          −          1                            x                              2                          n              −              1                                −          1                            1      +              x                              2                          n              −              1                                            →            2              n        −        1              2     for   x  ↑  1.Since this holds for arbitrary   n  ∈      N  , we conclude   ‖  f      ‖    sup    =  ∞Hence, f is not bounded on   (  0  ,  1  ), so that the series is not uniformly convergent on   (  0  ,  1  )

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

Answered question

Answer & Explanation

New Questions in Precalculus