query: How can one prove that $\lim_{n\to\infty}(1 +\frac k n)^n = e^k$?
positive_passages.docid: 149296
positive_passages.text: Is it trivial to say $\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k}$
negative_passages.docid: 78116
negative_passages.text: What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists?
negative_passages.docid: 75268
negative_passages.text: Proving that $\lim_{n\rightarrow \infty} \frac{n^k}{2^n}=0$
negative_passages.docid: 122820
negative_passages.text: Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$
negative_passages.docid: 134787
negative_passages.text: How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
negative_passages.docid: 5436
negative_passages.text: How to prove $\lim \limits_{x \to 1^-} \sum\limits_{n=0}^\infty (-1)^nx^{n²} = \frac{1}{2} \ $?
negative_passages.docid: 149068
negative_passages.text: Prove: Assume $\lim_{n\to\infty}(a_n)=a, \lim_{n\to\infty}(b_n)=b$ then $\lim_{n\to\infty}(a_n+b_n)=a+b$
negative_passages.docid: 128128
negative_passages.text: Show that $\lim_{n \to \infty} \left(1+\frac{a_n}{n}\right)^n= e^a$ when $a_n \to a$
negative_passages.docid: 21380
negative_passages.text: Proof for $e^z = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x$
negative_passages.docid: 144310
negative_passages.text: How would you prove that $\lim\limits_{n\to\infty}(\sqrt{4n^2+n}-2n)=\frac14$?
negative_passages.docid: 7510
negative_passages.text: Methods to find $\lim\limits_{n\to\infty}\frac1n\sum\limits_{k=1}^nn^{1/k} $
negative_passages.docid: 125860
negative_passages.text: Showing that $\lim\limits_{n\to\infty}x_n$ exists, where $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$
negative_passages.docid: 5728
negative_passages.text: How to prove $\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n} - 1) = 0$?
negative_passages.docid: 143227
negative_passages.text: Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
negative_passages.docid: 155333
negative_passages.text: Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
negative_passages.docid: 76756
negative_passages.text: A convergence proof: $\lim_{n\to\infty} \left(1+n^2\right)^{\frac1n}$
negative_passages.docid: 16143
negative_passages.text: How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?