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If a partition of an integral diverges, does the whole integral diverge?

  1. Sep 16, 2012 #1
    [tex] \int^{b}_{a}f(x)dx = \int^{c}_{a}f(x)dx + \int^{b}_{c}f(x)dx [/tex]

    If one of the integrals on the right-hand-side is known to diverge, must the integral on the left also necessarily diverge?

  2. jcsd
  3. Sep 16, 2012 #2
    Yes, by definition of improper integrals.
  4. Sep 24, 2012 #3
    Yes if c is between a and b. But not necessarily if c is not between a and b:

    [tex] \int_{1}^{2} \frac{1}{x}dx [/tex] converges, but [tex] \int_{1}^{-1}\frac{1}{x}dx + \int_{-1}^{2} \frac{1}{x}dx [/tex] diverges (each term is an integral over a region containing 0, where 1/x is unbounded).
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