1. Muammo: Berilgan qatorni yaqinlashishga tekshirish: $$\sum_{n=2}^{\infty} \frac{1}{(n+2) \ln^{2} n}$$. 2. Qatorning umumiy hadining ifodasi: $$a_n = \frac{1}{(n+2) \ln^{2} n}$$. 3. Qatorning yaqinlashishini tekshirish uchun integral testi qo'llaniladi, chunki $a_n$ musbat, kamayuvchi va $f(x) = \frac{1}{(x+2) \ln^{2} x}$ funksiya sifatida uzluksiz, kamayuvchi va musbat $x \geq 2$ uchun. 4. Integral testi: agar $$\int_2^{\infty} \frac{1}{(x+2) \ln^{2} x} dx$$ yaqinlashsa, qator ham yaqinlashadi. 5. $x+2 \sim x$ katta $x$ uchun, shuning uchun integralni taxminan $$\int_2^{\infty} \frac{1}{x \ln^{2} x} dx$$ sifatida ko'rib chiqamiz. 6. O'zgartirish: $t = \ln x$, shunda $dt = \frac{1}{x} dx$, integral quyidagicha bo'ladi: $$\int_{\ln 2}^{\infty} \frac{1}{t^{2}} dt$$. 7. Bu integralni hisoblaymiz: $$\int_{\ln 2}^{\infty} t^{-2} dt = \left[-t^{-1}\right]_{\ln 2}^{\infty} = 0 - \left(-\frac{1}{\ln 2}\right) = \frac{1}{\ln 2}$$. 8. Integral yaqinlashadi, demak qator ham yaqinlashadi. Javob: Qator $$\sum_{n=2}^{\infty} \frac{1}{(n+2) \ln^{2} n}$$ yaqinlashadi.