B Integral test and its conclusion

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1. Aug 3, 2016

The Subject

When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?

2. Aug 3, 2016

Stephen Tashi

Look at an example. What value do you get from $$\int_0^\infty 3x\ dx$$ ?

3. Aug 3, 2016

The Subject

ok so I get
$$\lim_{t \to \infty} \int_0^t 3x dx = \lim_{t \to \infty} \frac{3}{2}x^2 |_0^t=lim_{t \to \infty} \bigg(\frac{3}{2}t^2 - \frac{3}{2}0^2\bigg)=\infty$$

4. Aug 3, 2016

Stephen Tashi

So what does the integral test say about the convergence or divergence of the infinite series $0 + 3 + 6 + 9 + 12 + ...$ ?

5. Aug 3, 2016

The Subject

The series 0 + 3 + ... diverges. Since f(x) div, an also diverges. I get it how to use it now.

Thanks!

6. Aug 3, 2016

Stephen Tashi

Sketch a function with a positive graph and, on top of that, sketch the rectangles whose areas represent the terms of the related series. These rectangles have bases [0,1], [1,2] ... etc. and heights determined by the function's value at the left endpoints. The area of the rectangles is not a particularly good approximation to the area under the graph, but the intuitive idea is that the two areas are either both finite or both infinite.