1. a) Discuss the convergence of exponential series $\sum_{n=0}^\infty \frac{x^n}{(n+1)!}$.\n\nStep 1: State the problem: Determine if the series $\sum_{n=0}^\infty \frac{x^n}{(n+1)!}$ converges for all real $x$.\n\nStep 2: Recall the ratio test for convergence: For a series $\sum a_n$, if $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L$, then the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L=1$.\n\nStep 3: Compute the ratio:\n$$\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{x^{n+1}}{(n+2)!}}{\frac{x^n}{(n+1)!}} \right| = \left| \frac{x^{n+1}}{(n+2)!} \cdot \frac{(n+1)!}{x^n} \right| = \left| \frac{x}{n+2} \right|$$\n\nStep 4: Take the limit as $n \to \infty$:\n$$\lim_{n \to \infty} \left| \frac{x}{n+2} \right| = 0$$\n\nStep 5: Since $0 < 1$, the series converges for all real $x$.\n\nAnswer: The exponential series $\sum_{n=0}^\infty \frac{x^n}{(n+1)!}$ converges for all real values of $x$.