Homework SourseLet fx the sum from n0 to infinity of n1xn a find the domai
Let:
f(x) = the sum from n=0 to infinity of [(n+1)x^n]
a) find the domain and obtain an explicit formula for the function f(x)
b) evaluate f(1/4)
f(x) = the sum from n=0 to infinity of [(n+1)x^n]
a) find the domain and obtain an explicit formula for the function f(x)
b) evaluate f(1/4)
Solution
4 conditions ? 4 unknowns ? cubic function f(x) = ax³ + bx² + cx + d f\'(x) = 3ax² + 2bx + c f(2) = 8a + 4b + 2c + d = 4 f(3) = 27a + 9b + 3c + d = 3 f\'(2) = 12a + 4b + c = -1 f\'(3) = 27a + 6b + c = -5 12a + 4b + c = -1 27a + 6b + c = -5 ? 15a + 2b = -4 19a + 5b + c = -1 12a + 4b + c = -1 ? 7a + b = 0 15a + 2b = -4 7a + b = 0 ? -14a - 2b = 0 ? a = -4 7a + b = 0 ? b = 28 12a + 4b + c = -1 ? 12*-4 + 4*28 + c = -1 ? -48 + 112 + c = -1 ? c = -65 8a + 4b + 2c + d = 4 ? 8*-4 + 4*28 + 2*-65 + d = 4 ? -32 + 112 - 130 + d = 4 ? d = 54 f(x) = -4x³ + 28x² - 65x + 54 (Note f\'(x) = -12x² + 56x - 65)![Let: f(x) = the sum from n=0 to infinity of [(n+1)x^n] a) find the domain and obtain an explicit formula for the function f(x) b) evaluate f(1/4)Solution 4 cond](/WebImages/46/let-fx-the-sum-from-n0-to-infinity-of-n1xn-a-find-the-domai-1144143-1761614426-0.webp "Let: f(x) = the sum from n=0 to infinity of [(n+1)x^n] a) find the domain and obtain an explicit formula for the function f(x) b) evaluate f(1/4)Solution 4 cond")
