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Showing the convergence of the given recursively defined sequence

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#real-analysis #sequences-and-series #recurrence-relations
2k
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Divide 56 in four parts in AP such that the ratio of product of their extremes 1st and 4th to product of means 2nd and 3rd is 5:6

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#sequences-and-series #arithmetic-progressions
1.1k
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If $\lim_{n \to \infty} \frac{x_{n+1}}{x_n} = a > 0$, prove that $\lim_{n \to \infty} \sqrt[n]{x_n} = a$

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#real-analysis #sequences-and-series #limits #convergence-divergence #proof-explanation
47
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Convergence of $\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$

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#real-analysis #sequences-and-series #convergence-divergence
114
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Conjecture on limit of a partial fraction expansion involving tan and cot

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#integration #sequences-and-series #limits #special-functions
210
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Ratio of 0's to 1's in the Fibonacci Word is the golden ratio

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#sequences-and-series #fibonacci-numbers #golden-ratio
373
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How to prove that $\sum_{k=1}^n\frac{1}{\sqrt[n]{k!} }\sim \frac{n}{\ln n}$

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#sequences-and-series #limits #summation
97
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Infinite size is actually number density?

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#sequences-and-series #infinity
156
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Convergence of $\sum_{n=1}^\infty \frac{1}{n^{a_n}}$

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#real-analysis #sequences-and-series #analysis #convergence-divergence
924
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Generating function for sequence of natural numbers?

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#sequences-and-series #recurrence-relations #generating-functions #natural-numbers
5.3k
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How to write the index number in the Wolfram Alpha search engine

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#sequences-and-series #wolfram-alpha
1.4k
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Completing the space of series so there is a slowest converging series

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#real-analysis #sequences-and-series #functional-analysis #asymptotics #nonstandard-analysis
60
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Are two sequences comparable with big O notation if they are positive, decreasing and with limit $0$?

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#sequences-and-series #asymptotics
79
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Does the unordered Look-And-Say sequence always loop?

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#sequences-and-series
72
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On the convergence of $\sum\limits_{n \geq 0} \frac{x_{n}}{n}$

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#sequences-and-series

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