"S=\\frac{1}{1!}+\\frac{1}{3!}+\\frac{1}{5!}+...+\\frac{1}{(2n+1)!}+..."
apply the d'Alembert ratio test to
"a_n=\\frac{1}{(2n+1)!},\\\\\na_{n+1}=\\frac{1}{(2n+3)!},\\\\"
consider
"lim_{n\\mapsto \\infty} \\frac{a_{n+1}}{a_n}=lim_{n\\mapsto \\infty} \\frac{(2n+1)!}{(2n+3)!}=\\\\\n=lim_{n\\mapsto \\infty} \\frac{(2n+1)!}{(2n+1)!(2n+2)(2n+3)}=\\\\\n=lim_{n\\mapsto \\infty} \\frac{1}{(2n+2)(2n+3)}=0<1."
Since the limit value 0 is less than 1, the series is convergent.
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