Does the following series converge or diverge?

Question

$$\sum_n\frac{2n-1}{n!}$$

I used the ratio test here and got the lim as $n \to \infty$ to be $0$. Therefore, I assumed that the series converges. However, my textbook says that it diverges.

How do I do this problem?

Answer 1

Your textbook is wrong. This series converges. Your reasoning is correct. (As a textbook author, I sympathize with the person who made the mistake; as a teacher, I sympathize with you as a student, and applaud you for thinking your reasoning was good!)

Answer 2

Any polynomial function grows slower than a factorial function, so the limit must converge.

Answer 3

As John has pointed out, your textbook is wrong: the series converges. In fact, it's easy to find its value. For example, $\sum _{n=1}^\infty \frac{2n-1}{n!} = 2 \sum _{n=1} ^\infty \frac{1}{(n-1)!} - \sum _{n=1}^\infty \frac{1}{n!} = 2e - e + 1 = e + 1$.

Answer 4

You can also see that

$$\sum_{n=1}^{\infty} \frac{2n-1}{n!}=\left(\sum_{n=1}^{\infty} \frac{x^{2n-1}}{n!}\right)'\left.\right|_{x=1}=\left(\frac{e^{x^2}-1}{x}\right)'\left.\right|_{x=1}=e+1$$