MAT 21C Lecture Notes - Lecture 12: Absolute Convergence, Ratio Test, Horse Length
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Series: a 1 + a 2 + a 3 + a 4 + l a n l means: Given > 0, there exists an n so that |a n - l| + whenever n >= n. And perhaps diverges for some other values of x. We look at the possible intervals of convergence : Taylor series: f(x) f(a) + f"(a)(x-a) + f""(a)(x-a) 2 /2! Under favorable conditions: f(x) = f(a) + f"(a)(x-a) + f""(a)(x-a) 2 /2! Also denoted as: f(x) = f(a) + f"(a)/1! The above is called a second order taylor polynomial + r n (x) When f(x) does not have a derivative (e. g. f(x) = |x|) When x is outside the radius of convergence: Example: e x 1 + x + x 2 /2!+ x 3 /3! This test tells us that it converges absolutely for any x. Now look at this using p 2 (x): e 1 = 1 + 1 + = 2. 5 =p 2 (1)