The mishna in Taanis asks when we should start saying mashiv haruach ["gvuros geshamim"] and offers two tannaic opinions [either from the first day of Succos or according to Rebbe Yehoshua until the last day of Succos - Shmini Atzeres]. The gemara [2a] asks how we know that we have to mention rain at all that prompted us to ask from when we begin. If it is because the mishna in brachos says that we praise Hashem for rain in the bracha of מחיה המתים then, by golly, put this mishna in Seder Zeraim with Brachos and not in Seder Moed. The gemara concludes that we are basing ourselves on the mishna in Rosh Hashna that says that on Succos we are judged on water.
The question however still irks us - put this mishna in Brachos after we talk about the obligation to praise Hashem for rain? [See Tosfos] That would be its natural place more than after mentioning the judgement on water.
Rashi writes that we have to "appease" [לרצות] Hashem when it is judgement time on the water by praising Him [see lashon Rashi in Taanis].
Kumpt ois - it emerges from Rashi that there are TWO obligations to praise Hashem or rain. There is a din klali - a general obligation during the entire rainy season. In addition, there is a specific obligation to begin that applies at the time of judgement on the rain. That is why it would not have sufficed to put the mishna in Seder Zeraim because then we would have only known the general din of mentioning rain and not the specific din of mentioning rain at judgement time which is unique to the moed of Succos. Hence, its appearance in Seder Moed where we learn both dinim in one shot.
So we have an explicit mishna which teaches us that according Rebbe Yehoshua the din on rain extends until Shmini Atzeres which he call יום טוב האחרון של חג. Meaning that the din goes through Succos all the way until the end. So how GORGEOUS is it that we continue saying Li-Dovid Hashem through Succos until and including Shmini Atzeres.
That is all Shittas Rashi.
Uber Tayyyyyyysfisssssssss - Taysfiss halt andrish. Tosfot holds differently. They say that since we are judged about rain on Succos it must be the period before it is supposed to rain and therefore now is the time to start mentioning rain [but not that the din is actually on Shmini Atzeres]. According to them, there is no reason that would compel us to continue saying Li-Dovid Hashem on Shmini Atzeres, as is the custom of nusach sfard not to say Li-Dovid on Shmini Atzeres.
פלאי פלאות ממש.
עפ"י ספר מרבה טובה עמ' קלו-ז מאת מו"ר הרה"ג ר' יעקב דוד המניק שליט"א
And after Yuntiff I have a treaaaaat for you from the same sefer. But for now, the sun will soon be setting on Givat Ze'ev Hachadasha - ועוד חזון למועד!!