# Find \$sum_{n=0}^inftyfrac1{2^ncdot n!}\$ and \$sum_{n=0}^inftyfrac1{4n+3}\$ Code Answer

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Consider the set of integers \$S = {2^0.0!, 2^1.1!, 2^2.2!, ldots }\$. What do we get when we sum its series of reciprocals? Answer the same question for set \$S’ ={3, 7, 11, 15, 19, ldots }\$.

I try to use the series, \$\$sum_{n = 1}^infty frac{1}{n!} = e – 1 text{and} sum_{n = 1}^infty frac{1}{2^n} = 1\$\$ Should I have to show that the series diverges or converges?

I get the original series, sum of reciprocals of \$S < 2e\$, so it should converge. I am not sure my answer is correct.

Hint:

Notice that for the first that \$\$e^{1/2} = sum_{n=0}^{infty}frac{1}{2^n n!}\$\$

Use the Comparison test to the second to see it diverges.

\$\$4n + 3 < 4n + 4\$\$