Math 341 Lecture #30 §6.4: Series of Functions

For each n ∈ N we havef n(x) ′ = −4nx3(n 4 + x 4 ) . 2For a real K > 0, we have for all x ∈ [−K, K] that∣ |f n(x)| ′ ≤4nx 3 ∣∣∣∣ ≤ 4nK3n 4 + x 4 n 8Since∞∑n=14K 3n 7= 4K3n 7 .converges, we have that f ′ n converges uniformly on [−K, K] by the Weierstrass M-Test.Since ∑ ∞n=1 f n(x 0 ) converges to f(x 0 ) for any x 0 ∈ [−K, K], then by Theorem 6.4.3, wehave that f(x) is differentiable on [−K, K] withf ′ (x) =∞∑f n(x) ′ =n=1∞∑n=1−4nx 3(n 4 + x 4 ) 2 .Since K > 0 is arbitrary, we have that f is differentiable on R.