For fixed positive numbers \( x_1, x_2, \dots, x_m \), we define a sequence \( \{ a_n \} \) by \( a_n = x_n \) for \(n \leq m \) and
\[
a_n = a_{n-1}^r + a_{n-2}^r + \dots + a_{n-k}^r
\]
for \( n > m \), where \( r \in (0, 1) \). Find \( \lim_{n \to \infty} a_n \).
Find the following limit:
\[
\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)
\]
For \( a \geq 0 \), find
\[
\lim_{n \to \infty} n \int_{-1}^0 \left( x + \frac{x^2}{2} + e^{ax} \right)^n dx.
\]
Does \(\frac{1}{n \sin n}\) converge as \(n\) goes to infinity?
For a nonnegative real number \(x\), let \[ f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}\] for a positive integer \(n\). Determine \(\lim_{n\to\infty}f_n(x)\).
For real numbers \( a, b \), find the following limit.
\[
\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.
\]
Let \(a_0=3\) and \(a_{n}=a_{n-1}+\sqrt{a_{n-1}^2+3}\) for all \(n\ge 1\). Determine \[\lim_{n\to\infty}\frac{a_n}{2^n}.\]
For given positive real numbers \(a_1,\ldots,a_k\) and for each integer n≥k, let \(a_{n+1}\) be the geometric mean of \( a_n, a_{n-1}, a_{n-2}, \ldots, a_{n-k+1}\). Prove that \( \lim_{n\to\infty} a_n\) exists and compute this limit.
Let \(f:\mathbb{R}^n\to \mathbb{R}^{n-1}\) be a function such that for each point a in \(\mathbb{R}^n\), the limit $$\lim_{x\to a} \frac{|f(x)-f(a)|}{|x-a|}$$ exists. Prove that f is a constant function.
Let f be a continuous function on [0,1]. Prove that \[ \lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).\]