asrcsin n+1 / n^2+1
_______________
(n+1)!
[m]lim_{n → ∞ }\frac{a_{n+1}}{a_{n}}=lim_{n → ∞ }\frac{\frac{arcsin \frac{n+1+1}{(n+1)^2+1}}{(n+1+1)!}}{\frac{arcsin \frac{n+1}{n^2+1}}{(n+1)!}}=[/m]
[m]=lim_{n → ∞ }\frac{arcsin \frac{n+2}{(n+1)^2+1}\cdot (n+1)!}{(n+2)!\cdot arcsin\frac{n+1}{n^2+1}}=lim_{n → ∞ }\frac{\frac{n+2}{((n+1)^2+1)(n+2)}}{\frac{n+1}{n^2+1}}=0 < 1[/m]
ряд сходится
[m]arcsinx ∼ x [/m] при [m]x → 0[/m]
[m]arcsin \frac{n+1}{n^2+1} ∼ \frac{n+1}{n^2+1}[/m] при [m]n → ∞ [/m]
[m]arcsin \frac{n+2}{(n+1)^2+1} ∼ \frac{n+2}{(n+1)^2+1}[/m] при [m]n → ∞ [/m]