Let A=∑n=0∞1n!A = \sum_{n=0}^{\infty} \frac{1}{n!}A=∑n=0∞n!1 and B=∑n=0∞(−1)nn!B = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!}B=∑n=0∞n!(−1)n. Let C=∑n=0∞cnC = \sum_{n=0}^{\infty} c_nC=∑n=0∞cn be their Cauchy product. Find cnc_ncn and the sum of the series CCC.
cn=0c_n = 0cn=0 for all n≥1n \ge 1n≥1, c0=1c_0 = 1c0=1, and the sum is 111.
cn=1+(−1)n2⋅n!c_n = \frac{1 + (-1)^n}{2 \cdot n!}cn=2⋅n!1+(−1)n, and the sum is cosh(1)\cosh(1)cosh(1).
cn=(−1)nn!c_n = \frac{(-1)^n}{n!}cn=n!(−1)n, and the sum is eee.
cn=2nn!c_n = \frac{2^n}{n!}cn=n!2n, and the sum is e2e^2e2.