Calculate LIM (a - 2 + a - 4 +...) +a⌒2n)/（a+a⌒2+a⌒3＋… +a⌒n）

Calculate LIM (a - 2 + a - 4 +...) +a⌒2n)/（a+a⌒2+a⌒3＋… +a⌒n）

(1) When a = 1, a ^ 2 + A ^ 4 + +a^2n=n,a+a^2+… +A ^ n = n, the original formula is 1
(2) When a ≠ 1, a ^ 2 + A ^ 4 + +a^2n=a²(1-a^2n)/(1-a²),a+a^2+a^3++a^n=a(1-a^n)/(1-a)
When the original formula = Lima (a ^ n + 1) / (a + 1) 1 | a | 1, the original formula = 1 | a | 1, it does not exist

Lobita's rule / / what's the meaning of LIM? What's the difference between LIM (f (x) / F (x)) and lim (f '(x) / F' (x))?
Let f (x) and f (x) satisfy the following conditions:
(1) When x → a, Lim f (x) = 0, Lim f (x) = 0;
(2) Both f (x) and f (x) are differentiable in a centreless neighborhood of point a, and the derivative of F (x) is not equal to 0;
(3) When x → a, LIM (f '(x) / F' (x)) exists or is infinite, then LIM (f (x) / F (x)) = LIM (f '(x) / F' (x)) when x → a

Lim means "limit". Limit theory is the foundation of higher mathematics. High school will learn the basic knowledge of limit
X - > a means that x tends to a (x is infinitely close to a)
LIM (f (x) / F (x)) denotes the limit of the ratio of two functions f (x) to f (x)
LIM (f '(x) / F' (x)) denotes the limit of the ratio of the derivative functions f '(x) and f' (x) of the above two functions
After the above concepts are clear, you can understand the lobita's law described below. It is a very important method for differential calculus to find limits

Power series (x ^ n) / (n + 1); n = 0, n tends to infinity; find the sum function s (x) in the interval (- 1,1)

Let f (x) = the power series, then f (x) = XF (x) = power series (x ^ n + 1) / (n + 1); n = 0, n tends to infinity, for the derivation of F (x): F '(x) = power series X ^ n; n = 0, n tends to infinity = 1 / (1-x); therefore, f (x) = - ln (1-x) + C, substituting f (0) = 0, C = 0, then XF (x) = - ln (1-x), that is:
xS(x)=-ln(1-x),x!=0;
S(x)=0,x=0.