Recent questions tagged limits

The series\[\sum_{n} \frac{3 \cdot 6 \cdot 9 \cdots 3 n}{7 \cdot 10 \cdot 13 \cdots(3 n+4)} x^{n}, \quad x>0\]converges for $0 < x \leq 1$ and diverges for $x 1$converge...

Given a real number $\alpha \in(0,1)$, define a sequence $\left\{x_{n}\right\}_{n \geq 0}$ by the following recurrence relation:\[x_{n+1}=\alpha x_{n}+(1-\alpha) x_{n-1},...

Let $S_{n}=\sum_{k=1}^{n} \frac{1}{k}$. Then$S_{2^{n}} \geq \frac{n}{2}$ for every $n \geq 1$$S_{n}$ is a bounded sequence.$\left|S_{2^{n}}-S_{2^{n-1}}\right| \rightarrow...