Relation between two series

Question

Consider the two series ,

A=Σ(2ⁿ/n!) from 1 to ∞.

and,

B=Σ(4ⁿ/n!) from 1 to ∞.

What is the relationship between them?( If any)

I think the exponential series might come in handy but the numerator is progressing geometrically.

I don't think squaring the series will help either since there are infinite terms.

I tried to form 2ⁿ somehow in the series B but couldn't do it since now the powers progressed in squares of 2.

Is there any way of finding a relation between the two? Or evaluating each series by anyway?

Answer 1

We know that, $$e^x = \sum_{n=0}^\infty\frac{x^n}{n!}=1+\sum_{n=1}^\infty\frac{x^n}{n!}$$ Hence, $$A = e^2-1$$ $$B = e^4-1$$ Now, you can see relation between the given series.

Hope it helps:)

Answer 2

Both series converge according to the Ratio Test. When you check $\frac {a_{n+1}}{a_n}$, you get $\frac {4}{n+1}$ and $\frac {2}{n+1}$ where both go to zero as $n$ goes to infinity.