What if the lecturer is not rigorous?

In physics we deliberately do not prove the series converges, because we are not interested in teaching concepts like convergence. Physics courses are not intended to be mathematically rigorous. It just is not one of the goals.

Let me add a thought following the other post, which asserts that this is not a mistake, pedagogically. Take this as a given: you don't need to "call out" the professor for failing to teach "properly." It is, however, still the case that you personally are wanting to dig into the mathematical foundations of these concepts more deeply, and find it important to your comprehension.

That's great! You might learn something really interesting, and might set yourself on a path to become a person who makes scientific advances by attacking these sorts of questions.

Now, I would suggest approaching your professor from that perspective, instead of considering it a problem with their teaching. Ask if there are books or other resources that the professor would suggest where you can learn more about the proofs behind these assertions. If the professor doesn't have good suggestions for you, try looking in places like Physics.SE. If you can't find a satisfactorily rigorous proof, it may well be that it does not exist (unlikely, but it happens), and that may be an interesting opportunity!

Echoing parts of the other answers, and some of the comments: first, it is inaccurate to declare omissions such as "proof of convergence" a "mistake". There simply is no absolute obligation to verify that all parts of the mathematics work as a physicist expects for other reasons. Yes, you or I and others might want to see the proof, that is, mathematical causality, but this is simply not obligatory. (Conversely, we can prove things without direct physical manifestations or physical reasoning...)

In fact, "convergence" is merely a simple form of what one might want, and itself not obligatory (much less its proof). Indeed, I have read that Poincare discovered in the late 19th century that a series expansion of a solution to a differential equation used for many decades (successfully) in celestial mechanics did not converge. Not that its converge was difficult to prove, but that it definitely diverged. But/and people had been getting correct numerical outcomes. Well, it was an "asymptotic expansion", ... but/and such expansions are more delicate in some regards (e.g., term-wise differentiation) than convergent power series, and the mathematical details were not filled in for several decades.

Another example is P.G.M. Dirac's book on quantum mechanics, which used distributions and unbounded operators in manners that would not be justified for 20 years (in the work of L. Schwartz). I have read that J. von Neumann and others were considerably disturbed by the lack of "rigor", or even the pretense of it, which motivated them to try to provide such... Nevertheless, the predictive and explanatory power of Dirac's work was unquestionable, and it would have been ridiculous to have dismissed it because he couldn't provide proofs, or didn't care to.

As remarked above, it really does appear to be that hard-to-justify mathematics is fairly tolerable when it quasi-magically predicts physical details, or quasi-magically proves to be an accurate book-keeping or computational device for observable physical phenomena.

Yes, we should think very differently when/if we aim to "subvert" such mathematics to purely mathematical situations, where there may be no genuine physical phenomenon to observe and test. No, I do not have that physics-y intuition that suggests (to my perception) outrageous mathematical manipulations, so I myself definitely need either or both pithy examples and persuasive (!) proofs that assure me there's some "causality" beyond the literal tangible world. But, in fact, history suggests that much interesting mathematics has come from "outrageous" mathematical stunts by imaginative physicists, so such stuff is a good source!

And, yes, sometimes the purely mathematical justification for obviously-necessary mathematical tricks in physics is far more sophisticated than the immediate physical explanation/motivation/phenomenon. Sure, sometimes the mathematics is not hard, and simply omitted due to lack of interest. Sometimes the mathematics is profoundly difficult, or in fact impossible in a particular year with technical limitations of the time. That fact, that has appeared over and over, is philosophically and scientifically provocative in itself, in my opinion.

So, yes, I, too, have been disturbed by reading physics-y accounts that did (to my perception) crazy mathematical things. Long ago, I thought that this was a definite failing, and that rigor was required, and possible. By now I see that these situations are much more complicated than that, and that gauging any particular instance may be unexpectedly non-trivial!