• Show that the infinite multiplication (1+1/1)(1+1/2)(1+1/3)... does not converge.
  • zkfcfbzr@lemmy.world
    English
    1·
    2 months ago
    solution

    The terms can be rewritten as:

    (2/1) * (3/2) * (4/3) * … * ((n+1)/n) * …

    Each numerator will cancel with the next denominator. In total everything cancels, so the answer is the empty product, 1.

    …Wait…

    Uhm, ignore that. Rather, consider the products we get when multiplying. We get: 2/1. 6/2. 24/6. Etc. That is, we have:

    Π (n = 1 to k) (n+1)/n = (k+1)! / k! = (k+1)k!/k! = k+1

    k+1 clearly goes to infinity as k → ∞, so our product diverges to infinity.