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Just a note

Rk can only be finite if higher terms in the series converge towards zero. However, it is not always immediately obvious whether a series must be convergent.
    For example,
  • x0 + x1 + x2 + x3 ... will only converge if x is less than one.

    Such as where x = 1/n.

    Similarly,
  • c0x0 + c1x1 + c2x2 + c3x3 ... will converge if cixi decreases with i.

    So if x = 2 and ci = 1/ i! then you end up with 1/1 + 2/1 + 4/2 + 8/6 + 16/24 + 32/20 + 64/720 ...
    which, after the 3rd term, is convergent.

    But if x = i! and ci = 1/2i = 2-i the series would be divergent, and Rk would be infinite.