Calculate (i.e. express without using infinite sum): $\frac{1}{1!\cdot1}+\frac{2}{2!\cdot2}+\frac{4}{4!\cdot4}+\frac{8}{6!\cdot8}+..$

Question

Calculate (i.e. express without using infinite sum):

$$\frac{1}{1!\cdot1}+\frac{2}{2!\cdot2}+\frac{4}{4!\cdot4}+\frac{8}{6!\cdot8}+..$$

In sum it would be:

$$\sum_{n=0}^{\infty}\frac{2^{n}}{(2n)!\cdot2^{n}} = \sum_{n=0}^{\infty}\frac{1}{(2n)!}$$

But now I somehow need to get rid off the sum symbol because the task clearly asks for an expression without it :P

But how can I remove the sum symbol? Two things come to my mind: Derivation and taylor-formula. But how to use derivation here if we got factorial... How use taylor if there isn't a point given (that one where you analyze at), and also it requires derivative as well... ^^

Please if you answer me please explain it simple I'm about frustrating because I don't understand this for several days and I'm absolutely sure it will be asked in the exam too.

Answer 1

This is $\cosh 1 = \frac{e^1+e^{-1}}{2} = \frac{e^2+1}{2e}$.

Answer 2

Hint: Your series looks a lot like $$\cosh(x)=1+\dfrac{x^2}{2!}+\dfrac{x^4}{4!}+\cdots$$