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I'm trying to calculate limit to this function and but I've not been able to figure out how to approach this.

The definition of sequence $S$ is

$S(1) = 3 $

And $ \forall \geq 2, S(n) = S(n-1) + \frac{3}{2(n!)} $

I need to calculate the following limit.

$\lim\limits_{n \to \infty} S(n)$

I have been able to prove that the sequence is monotonously increasing and is bounded also. But I'm having difficulty in calculating limit. Can anyone explain me how I should approach this problem?

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    $\begingroup$ Do you know that $e = 1+1+1/2+1/6+...+1/n!+...$? $\endgroup$ Aug 13, 2015 at 1:18
  • $\begingroup$ Yeah, I knew that. I expanded the left side like 10 times, but it didn't occur to me that it's resulting in series expansion of $e$. $\endgroup$ Aug 13, 2015 at 1:23
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    $\begingroup$ That's what experience is good for. $\endgroup$ Aug 13, 2015 at 1:23

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So you want $3+\frac{3}{2}(\frac{1}{2!}+ \frac1{3!}+\frac{1}{4!}+\dots)=3+\frac{3}{2}(e-2)=3+\frac{3e}{2}-3=\frac{3e}{2}$

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    $\begingroup$ I was expanding the sequence and then I totally failed to observe that I was getting expansion of $e$. Thanks @dREaM. $\endgroup$ Aug 13, 2015 at 1:18

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