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| <math>f(x) = \ln (5-x) \,</math><br /><br /> | |||
| <math>\frac{d}{dx} f(x) = -\frac{1}{5-x} = -\frac{1}{1-(x-4)} = -\sum_{n=0}^\infty (x-4)^n \,</math><br /><br /> | |||
| <math>\int df(x) = \int\left(-\sum_{n=0}^\infty (x-4)^n\right)dx \,</math><br /><br /> | |||
| <math>f(x) = C-\sum_{n=0}^\infty \frac{(x-4)^{n+1}}{n+1} \,</math><br /><br /> | |||
| <math>f(x) = C-\sum_{n=1}^\infty \frac{(x-4)^n}{n} \,</math><br /><br /> | |||
| <math>\ln (5-x) = C-\sum_{n=1}^\infty \frac{(x-4)^n}{n} \,</math><br /><br /> | |||
| <math>\ln (5-4) = C-\sum_{n=1}^\infty \frac{(4-4)^n}{n} \,</math><br /><br /> | |||
| <math>\ln (1) = C-\sum_{n=1}^\infty \frac{(0)^n}{n} \,</math><br /><br /> | |||
| <math>\ln (1) = C \,</math><br /><br /> | |||
| <math>C = 0 \,</math><br /><br /> | |||
| <math>f(x) = -\sum_{n=1}^\infty \frac{(x-4)^n}{n} \,</math><br /><br /> | |||