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While evaluating indefinite integral $\int\frac{e^x}{x}dx$ as an infinite series

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \implies \frac{e^x}{x} = \sum_{n=0}^{\infty} \frac{x^{n-1}}{n!} \implies \int \frac{e^x}{x} dx = \sum_{n=1}^{\infty} \frac{x^{n}}{n! \times n} + C$$

But then, my answer sheet says,$$\int \frac{e^x}{x} dx = \ln[x] + \sum_{n=1}^{\infty} \frac{x^{n}}{n! \times n} + C$$

I have no idea where this $\ln[x]$ comes from... Thanks in advance for help!

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2 Answers

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When $n=0$,$$ \frac{x^{n-1}}{n!}=x^{-1} $$So integrating gives the log.

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$$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1+x+x^2/2! + x^3/3! + \cdots$$

$$\frac{e^x}{x} = \sum_{n=0}^\infty \frac{x^{n-1}}{n!} = 1/x+1+x/2! + x^2/3! + \cdots$$

$$\int \frac{e^x}{x} dx = \int \frac{dx}{x} +\int dx (1+x/2! + x^2/3! + \cdots)$$

$$\int \frac{e^x}{x} dx = \ln |x| + x + \frac{x^2}{2\cdot 2!} + \frac{x^3}{3\cdot 3!} + \cdots + \textrm{const.}$$

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