1. **Problem:** Find the limit of the sequence $$a_n = \frac{2n^n}{(n+1)^n}$$ as $n \to \infty$. 2. **Formula and rules:** We use the property that $$\left(\frac{n}{n+1}\right)^n = \left(1 - \frac{1}{n+1}\right)^n$$ and recall that $$\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = \frac{1}{e}$$. 3. **Work:** $$a_n = 2 \cdot \left(\frac{n}{n+1}\right)^n = 2 \cdot \left(1 - \frac{1}{n+1}\right)^n$$ As $n \to \infty$, $$\left(1 - \frac{1}{n+1}\right)^n \to e^{-1}$$. 4. **Conclusion:** $$\lim_{n \to \infty} a_n = 2 \cdot e^{-1} = \frac{2}{e}$$. 1. **Problem:** Find the limit of the sequence $$b_n = \frac{a n^4 + n^3}{3 n^3 + 4 n}$$ where $a \in \mathbb{R}$. 2. **Formula and rules:** For rational functions of $n$, divide numerator and denominator by the highest power of $n$ in the denominator to find the limit. 3. **Work:** Divide numerator and denominator by $n^3$: $$b_n = \frac{a n^4 / n^3 + n^3 / n^3}{3 n^3 / n^3 + 4 n / n^3} = \frac{a n + 1}{3 + \frac{4}{n^2}}$$ As $n \to \infty$, $\frac{4}{n^2} \to 0$. - If $a \neq 0$, then $a n \to \pm \infty$ depending on the sign of $a$, so $b_n \to \pm \infty$. - If $a = 0$, then $$b_n = \frac{1}{3 + 0} = \frac{1}{3}$$. 4. **Conclusion:** - For $a = 0$, $$\lim_{n \to \infty} b_n = \frac{1}{3}$$. - For $a \neq 0$, the limit does not exist (diverges to infinity). 1. **Problem:** Find the limit of the sequence defined recursively by $$c_n = \frac{c_{n-1}}{2}$$ with initial values $c_1 = 1$ and $c_1 = 4$. 2. **Formula and rules:** This is a geometric sequence with ratio $\frac{1}{2}$. 3. **Work:** - For $c_1 = 1$: $$c_2 = \frac{1}{2}, c_3 = \frac{1}{4}, c_4 = \frac{1}{8}, \ldots$$ - For $c_1 = 4$: $$c_2 = 2, c_3 = 1, c_4 = \frac{1}{2}, \ldots$$ In both cases, the sequence tends to zero as $n \to \infty$. 4. **Conclusion:** $$\lim_{n \to \infty} c_n = 0$$ for both initial values. **Final answers:** - $$\lim_{n \to \infty} a_n = \frac{2}{e}$$ - $$\lim_{n \to \infty} b_n = \begin{cases} \frac{1}{3}, & a=0 \\ \text{does not exist}, & a \neq 0 \end{cases}$$ - $$\lim_{n \to \infty} c_n = 0$$ for both $c_1=1$ and $c_1=4$.