Are these equal?

  • Thread starter romsofia
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  • #1
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Main Question or Discussion Point

[tex]{\lim_{n \to \infty} \sum_{i}^{n} f(x_i)Δ x= \int^a_b f(x)\,dx= \sum_{i}^{\infty} f(x_i)Δx}[/tex]

I don't believe they are but I may be wrong.

Thanks for any help.
 
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  • #2
427
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Updated thread, wasn't as clear before.
 
  • #3
chiro
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[tex]{\lim_{n \to \infty} \sum_{i}^{n} f(x_i)Δ x= \int^a_b f(x)\,dx= \sum_{i}^{\infty} f(x_i)Δx}[/tex]

I don't believe they are but I may be wrong.

Thanks for any help.
The key thing with this is that the "dx" (your triangle x term) also goes to zero and should also be based on your limiting term (in your case the n term).

If your triangle x term does not go to zero as the number of terms goes to infinity, then you'll get nonsensical results.

If you want a better idea of why this happens consider the fact that f'(x) = [f(x+h)-f(x)]/h as h -> 0 in the form of a limit. Now we know that h x f'(x) ~ f(x+h)-f(x). Then if you add up all these h x f'(x) terms, you get your identity you described above.

If you need more information look up information on the Riemann Integral.
 

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