1. **State the problem:** Express the given series in summation notation and find the general term. 2. **Identify the sequences:** - Numerators form an arithmetic sequence: $1, 2, 3, \ldots, n+1$ with first term $a=1$ and common difference $d=1$. - Denominators form an arithmetic sequence: $n, n+1, n+2, \ldots, 2n$ with first term $a=n$ and common difference $d=1$. 3. **Find the $k$th term of numerator:** $$a_k = a + (k-1)d = 1 + (k-1) \times 1 = k$$ 4. **Find the $k$th term of denominator:** $$a_k = n + (k-1) \times 1 = n + k - 1$$ 5. **Write the $k$th term of the series:** $$T_k = \frac{k}{n + k - 1}$$ 6. **Express the series in summation notation:** $$S = \sum_{k=1}^{n+1} \frac{k}{n + k - 1}$$ 7. **Summary:** The series is the sum of terms $\frac{k}{n+k-1}$ from $k=1$ to $k=n+1$. **Final answer:** $$\boxed{S = \sum_{k=1}^{n+1} \frac{k}{n + k - 1}}$$