Are Limits, Integrals, & Series Equal?

[romsofia]

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

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

Thanks for any help.

 

[romsofia]

Updated thread, wasn't as clear before.

 

[chiro]

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

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.