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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.

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2 Answers 2

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

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  • $\begingroup$ But how can I know that in the exam? We aren't allowed to use anything. Does that mean I need to keep some special series in my mind? I don't see another way to get to that solution. $\endgroup$
    – tenepolis
    Sep 19, 2016 at 10:26
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    $\begingroup$ You need to know $e^x$ at a minimum. You can then spot this is the even-power terms at $x=1$. To delete the odd-power terms of any series $f\left( x\right)$, just use $\frac{f\left( x\right)+f\left( -x\right)}{2}$.(Similarly, $\frac{f\left( x\right)+f\left( -x\right)}{2}$ deletes the even-power terms.) $\endgroup$
    – J.G.
    Sep 19, 2016 at 10:29
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Hint: Your series looks a lot like $$\cosh(x)=1+\dfrac{x^2}{2!}+\dfrac{x^4}{4!}+\cdots$$

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  • $\begingroup$ I will ask the same question; Is there no way to get to that solution by doing calculation on paper? For me it sounds like I need to keep some / several well-known series in my mind in order to solve this task. Let's say I don't know that the thing in the task is similar to $\cosh$, what can I do? $\endgroup$
    – tenepolis
    Sep 19, 2016 at 10:27
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    $\begingroup$ I don't see a way to solve this if you don't know that $e^x=\sum_{k\geq 0}\dfrac{x^k}{k!}$ for instance. And yes you should remember the main Taylor series: exp, sin, cos, $(1+x)^{\alpha}$ and $\ln(1+x)$. $\endgroup$
    – Olivier Moschetta
    Sep 19, 2016 at 10:30

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