I would like to know if it is true that if one term (summand) diverges then the whole improper integral diverges. I would have thought that it would be like splitting a fraction, that divergance of the numerator can/may be compensated by the denominator, so it tells you nothing, that i would have to check if the other is finite.
If i split an improper integral:
$$\int_a^b fdx = \int_a^b g+h dx = \int_a^b g dx + \int_a^bh dx$$
then the Prof. says that if i can show that $\int_a^b gdx$ diverges then i dont even have to check the other summand.
(I realize that the statement would be true if we were considering absolute values but the Prof. did not mention absolute value/convergence)
It is true under extra conditions. For instance that the summands are non-negative. If it would be true without conditions then every integral would diverge.
Let it be that $\int fdx$ diverges. Then $\int gdx=\int\left(g-f\right)+fdx$ would diverge too.