# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Prove $\sum_{n=1}^{\infty}\frac{n(3n−2)}{n!}=4e$

How to prove the following ? $$\sum_{n=1}^{\infty}\frac{n(3n−2)}{n!}=4e$$ I do know the series expansion for $e^x$.You may use it...
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### The sequence of nubers a1,a2,…an is defined as [closed]

The sequence of nubers a1,a2,...an is defined as an=(1/n+1)-(1/n=2) for each integer n>=1.what is the sum of he first 15 term of this sequence? a. 1/272 b. 1/6 c. 7/16 d. 1/2 e. 15/34 And explain ...
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### The size of sets of positive integers not having distinct subsets with equal size and sum

Let us call a set $S$ of positive integers "good" if there does not exist a pair of distinct subsets $A,B\subseteq S$ who have equal size and an equal sum. Equivalently, a set $S$ is good if the ...
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### convergence of power series, expanded by maclaurin

I'm getting ready for a test and I stumbled upon a question which goes like this: A function f(x) is given and I need to expand it to a power series using Mcloren sequences and then calculate its area ...
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### Comparison test for alternating series

Give $\sum_n (-1)^n a_n$ where $a_n \geq 0$, if we have that $$0\leq c_n \leq a_n \leq b_n,$$ $c_n, b_n$ are monotone decreasing and go to zero, thus $\sum_n (-1)^n c_n$ and $\sum_n (-1)^n b_n$ both ...
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### set-theoretic definition of sum of sequences, difference of sequences, product of sequences,…

I have $f,g \in \Bbb R^\Bbb N$, I define: $(f+g):=\{(y,z)| z= f(y)+g(y) \}$ $(-f):=\{(y,z)| z=-f(y) \}$ $(f-g):=f+(-g)$ $(f\cdot g):=\{(y,z)| z= f(y)\cdot g(y) \}$ Is it correct? I have problem ...
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### Is the partial sum of cosine bounded?

Is it true that $\sum_{k=1}^n \cos k$ is a bounded sequence? If so how to prove? I want to prove the series of $\cos n/(\sqrt n)$ is convergent by abel test but I dont know if the partial sums of ...
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### How to determine sum of an alternating power series and to prove that sum is positive

I am working on a problem involving an alternating power series as follows: $$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$ $a$ and $b$ is constant with $0<x<1$ I would like to ...
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### Fractal dimension of the function $f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}$

Consider the function $$f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}\, .$$ This is a bizarre and fascinating function. A few properties of this function that SEEM to be true: ...
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### Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets

Okay so I have a measurable function $f$ and a set $E_{n,i}$ $$E_{n,i}=\left\{ x:\frac{i-1}{2^{n}} \leq f(x)<\frac{i}{2^{n}}\right\}$$ where $i=1,...,n2^{n}$ and $n=1,2,...$ Then I have another ...
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### Explicit solutions to a digamma function equation

My main question: Can we obtain the exact solutions from the following equation? $$\sum_{k=1}^{n}\cfrac{1}{k-x-1}=0$$ Notation: This problem was reached from the digamma function $\psi$ as ...
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### Deriving the sequence if the generating function is irreducible?

I am trying to better understand generating functions and how they can be derived / manipulated / etc. Right now I am operating on this identity, slightly modified from the answer here: For a ...
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### Positive function(or series) that tends to 0 at infinity

Consider these two questions: -If we have an ultimately positive function that tends to 0 at infinity, does this imply that the function has to be decreasing? -Similar questions but for series: if ...
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### Convergence of sequence $a_0 := b$ and $a_{n+1} = 2^{a_n}$ in $\hat{\mathbb{Z}}$.

I have a question about the convergence properties of a sequence in $\hat{\mathbb{Z}}$, the completion of $\mathbb{Z}$. It is part of an exercise is due to this syllabus. I got confused somewhere. ...
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### On a second set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of these claims and show where were my mistakes or inaccurancies? Also ...
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### On a first set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of this claims and show where were my mistakes or inaccurancies? Also ...
Consider recurrent formula for a sequence of numbers $(y_n)$ (either real or complex): $$a_k y_{n+k}+a_{k-1}y_{n+k-1}+\cdots+a_0y_n=\sum_{i=0}^k a_i y_{n + i} = 0$$ It's known that the exact explicit ...