泰勒 5^x

解答

泰勒

5^{x}

1 + ln (5) x + \frac{ln ^{2} ( 5 )}{2} x^{2} + \frac{ln ^{3} ( 5 )}{6} x^{3} + \frac{ln ^{4} ( 5 )}{24} x^{4} + \dots

求解步骤

应用泰勒公式

= 1 + \frac{\frac{d}{d x} ( 5 ^{x} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( 5 ^{x} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( 5 ^{x} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 1 + \frac{ln ( 5 )}{1 !} x + \frac{ln ^{2} ( 5 )}{2 !} x^{2} + \frac{ln ^{3} ( 5 )}{3 !} x^{3} + \frac{ln ^{4} ( 5 )}{4 !} x^{4} + \dots

整理后得

= 1 + ln (5) x + \frac{ln ^{2} ( 5 )}{2} x^{2} + \frac{ln ^{3} ( 5 )}{6} x^{3} + \frac{ln ^{4} ( 5 )}{24} x^{4} + \dots