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

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.

Answer

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.

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

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