query: To prove that a function is linear or two series are convergent
positive_passages.docid: 155475
positive_passages.text: $\lim\limits_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0$ implies $(a_n)_n,(b_n)_n$ are convergent
negative_passages.docid: 36849
negative_passages.text: Show convergence of a given series and find the limit.
negative_passages.docid: 136421
negative_passages.text: How to prove convergence of $\sum\limits_{n=1}^\infty \frac{\sqrt a_n}{n^\alpha}$
negative_passages.docid: 47922
negative_passages.text: Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ is convergent in $\mathbb{R}$
negative_passages.docid: 83040
negative_passages.text: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
negative_passages.docid: 91785
negative_passages.text: Test the uniform convergency of a series of function
negative_passages.docid: 143439
negative_passages.text: Prove that the series $\sum_{n =2}^{\infty} \frac{1}{(\ln n)^{\ln n}}$ converges.
negative_passages.docid: 69348
negative_passages.text: Prove that $\int \limits _{0}^{\infty}\frac{e^{-2x}-e^{-ax}}{x}\text{d}x$ converges for any $a>0$
negative_passages.docid: 90936
negative_passages.text: how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?
negative_passages.docid: 87834
negative_passages.text: How can we show that the following sequence converges?
negative_passages.docid: 121558
negative_passages.text: Showing that $n$ exponential functions are linearly independent.
negative_passages.docid: 29836
negative_passages.text: Prove: Convergent sequences are bounded
negative_passages.docid: 36243
negative_passages.text: Several ways to prove that $\sum\limits^\infty_{n=1}\left(1-\frac1{\sqrt{n}}\right)^n$ converges
negative_passages.docid: 133451
negative_passages.text: To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$
negative_passages.docid: 48500
negative_passages.text: Proof of convergence of sequences
negative_passages.docid: 69254
negative_passages.text: Which of the following series is uniformly convergent?
negative_passages.docid: 111353
negative_passages.text: Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges