<s>[INST] Express $1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$ in a simplifed form I need to express $$1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb  + \frac{1}{n + 1}\binom{n}{n}$$ in a simplified form.
So I used the identity $$(1+x)^n=1 + \binom{n}{1}x + \binom{n}{2}x^2 + \dotsb   + \binom{n}{n}x^n$$
Now on integrating both sides and putting $x=1$.
I am getting $$\frac{2^{n+1}}{n+1}$$ is equal to the given expression.But the answer in my book is  $$\frac{2^{n+1}-1}{n+1}.$$
Where does that -1 term in the numerator come from?
 [/INST] Among $n+1$ people, a subset is randomly selected (i.e., each person will be in the subset or not with probability $1/2$). Then one person in the subset (if it is nonempty) is selected at random to win a prize. What's the probability that I (one of the $n+1$ people) win it?
There are $\binom{n}{k}$ ways to pick a subset of size $k+1$ that contains me; the probability of that subset is $\frac{1}{2^{n+1}}$, and the probability that I am the one selected is $\frac{1}{k+1}$. So the desired probability is
$$
\frac{1}{2^{n+1}} \sum_{k=0}^n \frac{1}{k+1} \binom{n}{k}. \tag{1}
$$
On the other hand, everyone out of the $n+1$ has an equal chance of winning, and there is only a $\frac{1}{2^{n+1}}$ chance of no one being selected, so the probability is
$$
\frac{2^{n+1} - 1}{2^{n+1}}  \cdot \frac{1}{n+1}. \tag{2}
$$
Thus (1) and (2) are equal, and if we multiply by $2^{n+1}$ we get
$$
\sum_{k=0}^n \frac{1}{k+1} \binom{n}{k}
= \frac{2^{n+1} - 1}{n+1}.
$$
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