query: Limit of a sequence defined by a sum: $\lim_{n\to \infty} \frac1{2^n}\sum_{k=1}^n \frac1{\sqrt k}\binom nk$
positive_passages.docid: 155382
positive_passages.text: Find $ \lim\limits_{{n \to \infty}} \frac1{2^n} \sum\limits_{k=1}^n \frac1{\sqrt{k}} \binom nk$
negative_passages.docid: 86965
negative_passages.text: Limit of this recursive sequence: $x_{n+1}=\bigl(1-\frac{1}{2n}\bigr)x_{n}+\frac{1}{2n}x_{n-1}.$
negative_passages.docid: 37059
negative_passages.text: limit of the sum $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n} $
negative_passages.docid: 21504
negative_passages.text: Showing that $\lim\limits_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$
negative_passages.docid: 128200
negative_passages.text: Sequence sum question: $\sum_{n=0}^{\infty}nk^n$
negative_passages.docid: 25507
negative_passages.text: Find the limit of $\sum\limits_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$
negative_passages.docid: 125538
negative_passages.text: limit of $n^{\frac{1}{\sqrt{n}}}$
negative_passages.docid: 24033
negative_passages.text: Infinite Series $\sum\limits_{k=1}^{\infty}\frac{k^n}{k!}$
negative_passages.docid: 51502
negative_passages.text: To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$
negative_passages.docid: 154553
negative_passages.text: If $a_k\geq0$ is a decreasing sequence converging to $0$, do we have $\lim_{n\to\infty}\frac{\sum_{k=1}^n a_k}n = 0$?
negative_passages.docid: 90773
negative_passages.text: $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k}{k^{2}+1}$
negative_passages.docid: 89263
negative_passages.text: Find the limit $\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$
negative_passages.docid: 37148
negative_passages.text: How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
negative_passages.docid: 49860
negative_passages.text: Determine the convergence of $\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{n(n-1)}$
negative_passages.docid: 152557
negative_passages.text: Calculate $ \lim\limits_{n \to \infty} \sum\limits_{k=0}^n \frac{2^{k-1}}{a^{2k-1}+1} $
negative_passages.docid: 28669
negative_passages.text: Find limit of the following sequence: $\frac{\log(n+1)}{\log(n)}$?
negative_passages.docid: 686
negative_passages.text: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$