Question on the Cauchy Condensation test

  1. Simfish

    Simfish 825
    Gold Member

    So it's here

    My question is this: is the value of the base 2 in 2^k an arbitrary value? Or is there something special about 2? Can we just use something like e^k instead?
  2. jcsd
  3. HallsofIvy

    HallsofIvy 41,264
    Staff Emeritus
    Science Advisor

    The 2 is not important, it's the exponential that is crucial.
  4. Notice that the function f in [itex]\sum_{n=0}^{\infty} 2^{n}f(2^{n})[/itex] need not be well-defined for all arguments in the real numbers, but only for the natural numbers (including the zero).
    What you have is a positive monotone decreasing sequence (which is denoted here by [itex]f(n)[/itex] but could just as well be written as [itex]a_n[/itex]). Now consider the sum (infinite series): the idea now is to form summation blocks with length [itex]2^n[/itex] and find a useful estimate for each block. Every block incorporates [itex]2^n[/itex] summands and the biggest summand in each block is the first one, as the sequence is monotone decreasing.

    Try writing this down and see what you end up with. If you can, check the converse.
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