Expressing sum using simple formula (without summation)

Question

Express by a simple formula not containing a sum:

$$\sum\limits^{n}_{k=1} \binom{k}{m}\frac{1}{k}$$

I figured that

$$\sum\limits^{n}_{k=1} \binom{k}{m}\frac{1}{k} = \frac{1}{k}\sum\limits^{n}_{k=1} \binom{k}{m}$$

So I could use $$\binom{m}{m} + \binom{m+1}{m} + \binom{m+2}{m} + \cdots + \binom{k}{m} = \binom{k+1}{m+1}$$

$$\sum\limits^{n}_{k=1} \binom{k}{m}\frac{1}{k} = \frac{1}{k} \binom{k+1}{m+1}$$

Is this derivation correct?

Answer 1

Notice that ${k\choose m}\frac{1}{k} = {k-1 \choose m-1}\frac{1}{m}.$ Then $$\sum\limits^{n}_{k=1} \binom{k}{m}\frac{1}{k}= \sum\limits^{n}_{k=1} \binom{k-1}{m-1}\frac{1}{m}.$$ Shifting the index, we can use the identity $\sum_{k=0}^{n-1}\binom{k}{m-1}=\binom{n}{m}.$ So $$\sum\limits^{n}_{k=1} \binom{k}{m}\frac{1}{k} = \frac{1}{m}\binom {n}{m}.$$