Limit with factorial and summation Finding $$\lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}$$   Try: $$\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}=\frac{n^3}{(n+1)(n+2)(n+3)}\sum^{n}_{k=0}(k+1)(k+2)\frac{\binom{n+3}{k+3}}{n^{k+3}}$$  Could some help me to solve it , thanks
Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} $ Evaluate $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}$.   $\bf{My\; Try::}$ Although we can solve it by converting into definite Integration.  But I want to solve it without Using Integration.  So $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} = \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{(n-1)(n-2).......(n-k+1)}{k!\cdot n^{k}\cdot (k+3)}$  Now How can i solve after that, Help required, Thanks
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They have been training for an upcoming, prearranged final battle between all the existing national governments; the military units are very possibly robotic or entirely digital in nature, so as to avoid wasting materiel / human life.    They ultimately win the battle, just barely; I think the girl protagonist gives them an edge in the competition by hacking the timing / control system in favor of their team, causing a discrepancy which only the boy protagonist notices due to his heightened perceptions.  It was agreed that the winning government gets dominion over one of the few islands of land left that was not irradiated in some previous war; the protagonists get to live there as a reward for their victory.  I believe the boy protagonist later realizes he's gradually losing his ability to exactly quantify the passage of time.  Published no later than 2004, likely somewhere between 1980-2000, probably sometime in the '90s. I've watched the Vampire Knight anime. 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