Здравствуйте, Alik4546.

f (x) = [$8721$]{k=0..+[$8734$]}((f^(k)(0) / k!) * x^k), при этом f⁰(x) = f(x), а 0! = 1.
f(0) = 1
f'(x) = 1/2*(1+x)^-1/2 [$8658$] f'(0) = 1/2
f''(x) = -1/4*(1+x)^-3/2 [$8658$] f'' (0) = -1/4
f'''(x) = 3/8*(1+x)^-5/2 [$8658$] f''' (0) = 3/8
f⁽⁴⁾(x) = -15/16*(1+x)^-7/2 [$8658$] f⁽⁴⁾ (0) = -15/16
f⁽⁵⁾(x) = 105/32*(1+x)^-9/2 [$8658$] f⁽⁵⁾ (0) = 105/32
...
f⁽ⁿ⁾(x) = (-1)ⁿ⁺¹*((2n-1)! / 2ⁿ⁺¹*(n-1)!) * (1+x)^-(2n-1)/2 [$8658$] f⁽ⁿ⁾ (0) = (-1)ⁿ⁺¹*(2n-1)! / 2ⁿ⁺¹*(n-1)!

Отсюда f(x) = 1 + [$8721$]{n=1..+[$8734$]}((-1)ⁿ⁺¹*(2n-1)! / 2ⁿ⁺¹*(n-1)!) * xⁿ/n! = 1 + [$8721$]{n=1..+[$8734$]} ((-1)ⁿ⁺¹*(2n)! / 4*2ⁿ*(n!)²) * xⁿ