The series $\sum\frac{(n+\sqrt{n})^{n}}{2^{n}n^{n+1}}$ is:

Question

Accepted Answer

To determine the convergence of the series $\sum \frac{(n+\sqrt{n})^n}{2^n n^{n+1}}$, we need to analyze the behavior of the general term $a_n = \frac{(n+\sqrt{n})^n}{2^n n^{n+1}}$ as $n$ approaches infinity.

### Explanation for Option 1: Convergent

1. **Simplification of the General Term:**
   - Consider the expression $(n+\sqrt{n})^n$. Using the binomial expansion, we have:
     $$
     (n+\sqrt{n})^n = n^n \left(1 + \frac{1}{\sqrt{n}}\right)^n
     $$
   - For large $n$, $\left(1 + \frac{1}{\sqrt{n}}\right)^n$ can be approximated using the limit:
     $$
     \left(1 + \frac{1}{\sqrt{n}}\right)^n \approx e^{\sqrt{n}}
     $$
   - Thus, the term becomes:
     $$
     a_n = \frac{n^n e^{\sqrt{n}}}{2^n n^{n+1}} = \frac{e^{\sqrt{n}}}{2^n n}
     $$

2. **Behavior of the Term:**
   - As $n$ becomes very large, $e^{\sqrt{n}}$ grows, but $2^n$ grows exponentially faster than $e^{\sqrt{n}}$.
   - Therefore, the term $\frac{e^{\sqrt{n}}}{2^n n}$ approaches zero as $n$ approaches infinity.

3. **Convergence Test:**
   - Since the terms $a_n$ approach zero and decrease rapidly due to the exponential growth of $2^n$, the series converges by the comparison test or the ratio test.

### Explanation for Other Options:

- **Option 2: Divergent**
  - This option is incorrect because the terms $a_n$ decrease to zero rapidly due to the dominance of $2^n$ in the denominator, ensuring convergence rather than divergence.

- **Option 3: Oscillatory**
  - This option is incorrect because the series does not exhibit any oscillatory behavior. The terms are positive and decrease monotonically to zero.

- **Option 4: None of these**
  - This option is incorrect because we have established that the series is indeed convergent, aligning with Option 1.

In conclusion, the series $\sum \frac{(n+\sqrt{n})^n}{2^n n^{n+1}}$ is convergent, and Option 1 is the correct choice. The rapid growth of $2^n$ in the denominator ensures that the terms decrease to zero, leading to convergence.