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

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Main Question or Discussion Point

[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?

BiP
 

Answers and Replies

  • #2
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Yes, by definition of improper integrals.
 
  • #3
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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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