Articles of sequences and series

Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”

I am doing some tests with strictly increasing integer sequences whose gaps between consecutive elements show a “pseudorandom” behavior, meaning “pseudorandom” that the gaps do not grow up continuously, but they change from a bigger value to a smaller one and vice versa due to the properties of the sequence without an easy way of […]

How to find the limit of the sequence $x_n =\frac{1}{2}$, if $x_0=0$ and $x_1=1$?

The formula is only applicable on values for $n\geq 2$. I know that the sequence is monotonic with a lower bound at $\frac 1 2$, but I am unsure how to find the supremum of the sequence. EDIT: $x_2 = \frac 1 2, x_3 = \frac 3 4, x_4 = \frac 5 8$. Does that […]

Find sum of series $\frac{1}{6} +\frac{5}{6\cdot12} +\frac{5\cdot8}{6\cdot12\cdot18} +\frac{5\cdot8\cdot11}{6\cdot12\cdot18\cdot24}+…$

How to find sum of above series $$\frac{1}{6} +\frac{5}{6\cdot12} +\frac{5\cdot8}{6\cdot12\cdot18} +\frac{5\cdot8\cdot11}{6\cdot12\cdot18\cdot24}+…$$ How to find sum of series I can find its convergence but not sum of series. Can anyone explain?

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828…$ We can rewrite this series as: $$Q=\sum_{k=0}^{\infty} a_k\tag{2}$$ $$a_k=\frac{\cos(\ln(2k+1))}{(2k+1)^{\epsilon}}-\frac{\cos(\ln(2k+2))}{(2k+2)^{\epsilon}}\tag{3}$$ Does this series converge? Using a method in a related question, we can show that (with $m=2k+1$) $$-a_k=m^{-1-\epsilon}\left(\epsilon\cos(\ln m)+\sin(\ln m)\right)+O(m^{-2-\epsilon}).\qquad m\to\infty \tag{4}$$ Thus $$|a_k|\le m^{-1-\epsilon}\left(\epsilon|\cos(\ln m)|+|\sin(\ln m)|\right)+O(m^{-2-\epsilon})=O(m^{-1-\epsilon})=O(k^{-1-\epsilon})\tag{5}$$ Therefore the series in (1) […]

Is $\frac{d}{dx}\left(\sum_{n = 0}^\infty x^n\right) = \sum_{n = 0}^\infty\left(\frac{d}{dx} x^n \right)$ true?

Almost 3 months ago, I asked this question regarding if it’s possible to compute the summation of derivatives, as in the example I’ve given: $$\sum_{n = 0}^\infty \frac{d}{dx} x^n$$ One answer regarded the interchange between summations and derivatives, which got me thinking: does the interchange between the derivative and the summation succeed in this example? […]

How did Euler prove the partial fraction expansion of the cotangent function: $\pi\cot(\pi z)=\frac1z+\sum_{k=1}^\infty(\frac1{z-k}+\frac1{z+k})$?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I’ve seen several modern proofs of it and they all seem to rely either on the Herglotz trick or on the residue theorem. I recon Euler had neither nor at his […]

Sequences of integers with lower density 0 and upper density 1.

It is possible to construct a sequence of integers with lower density 0 and upper density 1? where lower and upper density means asymptotic lower and upper density (cf. References on density of subsets of $\mathbb{N}$) EDIT: So, if this is true, then one can split $\mathbb{N}$ into to sequeneces of null lower density. I […]

Fibonacci General Formula – Is it obvious that the general term is an integer?

This question already has an answer here: How to prove that Fibonacci number is integer? [duplicate] 1 answer

Evaluating $\sum\limits_{n=0}^{20} \frac{(-1)^{n}2^{n+1}}{3^{n}},$

I know the sum of the series $$2 – \frac{4}{3} + \frac{8}{9} – \cdots + \frac{(-1)^{20}2^{21}}{3^{20}}$$ is equal to $$\sum\limits_{n=0}^{20} \frac{(-1)^{n}2^{n+1}}{3^{n}},$$ but I don’t know how to calculate the sum without manually entering it into the calculator.

Given $\sum |a_n|^2$ converges and $a_n \neq -1$, show that $\prod (1+a_n)$ converges to a non-zero limit implies $\sum a_n$ converges.

I have been working on this problem for a while and cannot seem to make any progress without coming up with something wrong or hitting a dead end. Here is what I have so far: $ \prod (1+a_n) &lt \infty \implies \sum a_n &lt \infty $: Similarly we ignore finitely many terms until $|a_n| \leq […]