Integral test and its conclusion

[The Subject]

I'm really confused about this test. Suppose we let f(n)=an and f(x) follows all the conditions.
When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?

 

[Stephen Tashi]

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

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

 

[The Subject]

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

 

[Stephen Tashi]

$$=\infty$$

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

 

[The Subject]

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

Thanks!

 

[Stephen Tashi]

Although I don't intuitively understand why this is true. .

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.