"I am not busy this week."
Could we say いそがしくない？ It sounds more natural for me, but I am not sure if it's correct
from what I have learned and understood so far, yes it's possible to say it, but 忙しくない is informal and less polite than 忙しくありません
if you were to compare 「忙しくないです」 and 「忙しくありません」, which one is more formal/polite or they are quite equivalent?
～ません is mainly used in written Japanese; ～ないです appears more often in spoken Japanese.
They're roughly equivalent, but there are some differences: https://japanese.stackexchange.com/q/2574/
According to the source cited by one answer:
masenhas a strong sense of formality, and often expresses the speaker's firm denial. Also, it is often used to add a sense of assertion or in emphatic constructions. This is thought to arise from the influence of the linguistic form (where the order is polite
n), in which the intent of negation appears stronger than the politeness. From this the speaker can draw the listener's attention to the negating function. If we assume that expressions which appear nearer to the end of the sentence leave stronger impressions, the negating function of
masenleaves a strong impression.
nai desuis often used in spoken Japanese. As indicated by Noda (2004), the effects of this construction are such that it is often used in concert with sentence-ending particles, and that it is easily joined with the
(verb) + shiteiruform. This is thought to arise from the influence of the linguistic form (where the order is negation
desu), in which the level of politeness appears stronger than the negating function. From this the speaker can draw the listener's attention to the level of politeness. If we assume that expressions which appear nearer to the end of the sentence leave stronger impressions, the politeness of
nai desuleaves a stronger impression. Also, it is further thought that this may have the effect of softening the speaker's strong assertion of a negative expression.
～ません and ～ないです are semantically equivalent, but ～ないです is softer and less insistent. If you need to give a firm denial with no wiggle room, go with ～ません. But since this level of bluntness can be inappropriate in some situations, ～ないです is there if you need it.
It isn't necessary. It just emphasizes the time as the topic of the sentence.
Why is は necessary? It will accept 今週はいそがしくないです, but not 今週いそがしくないです。This seems strange to me, because in other sentences like 今月いそがしくないです dropping the は was fine.
It does seem that there are inconsistencies. Strictly, "this week" is the topic so the topic particle is used to mark that. However, natural speech is another matter all together.
Japanese verbal variations are extensive, and sometimes while learning it can serve more in confusion when swapping between formal and informal literally from one test window to the next. Typical real life scenarios would see ourself and our interlocutor in the one mode for the conversation duration. It would be nice if occasionally there were categories here and there through the course which concentrated on formal only and informal only to aid knowlede consolidation. I feel particularly for those who are here to start from scratch where they will not have any kind of rule book knowledge available which is in the "traditional" courses. Like many here, I have studied elsewhere and know without the previously gained grammar rules, I would be making mistakes far more and more often.
I don't remember seeing the "く" in "いそがしく" before. Am I wrong, or what does it mean?
忙しい is an i-adjective. to make it into a negative sentence you first remove the trailing い from the i-adjective and then attach くない, thus the presence of く in 忙しくない.
But isn't the verb ある／あります used for inanimate objects? Then why is it used here?