How to restore a row from a function?

Question

Mathematics

They gave me a task - to restore in a general form the formula of a power series calculated in these functions. (in the form in which you would write a formula for math analysis). But here I am not in the tooth with my foot. Thank you.

1
for (s=0, sn = x, n=1; fabs(sn) > eps; n++)
{ s += sn;
sn= - sn * x / n; }

2
for (s=0, sn = x, n=2; fabs(sn) > eps; n+=2)
{ s += sn;
sn= sn * x *x * (n-1) / (n+1); }

Answer 1

https://en.wikipedia.org/wiki/%D0%A0%D1%8F%D0%B4_%...
1. e^(-x)
2. arth(x)

Answer 2

1
to make it completely clear, expand the series according to the algorithm
n = 1: x; n=2 : x^2/(1*2); n= 3 : x^3/(1*2*3*4); n=4 : x^4/(1*2*3*4*5*6) etc.