# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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### Euler sum symmetry formula

The following identity $$\sum_{n=1}^{\infty} \frac{\mathcal{H}_n^{(p)}}{n^q} + \sum_{n=1}^{\infty} \frac{\mathcal{H}_n^{(q)}}{n^p} = \frac{\zeta(p+q)+ \zeta(q) \zeta(p)}{2}$$ is considered obvious. ...
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### Longest geometric progression of primes

There are arbitrarily long arithmetic progressions of primes e.g. $5, 11, 17, 23, 29$ for a $5$-length progression, but no (infinite) arithmetic sequence of primes with common difference $d\neq 0$, as ...
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### How many binary arrays that have no $3$ consecutive $1$'s are there? [duplicate]

Let $N$ be a positive integer. The function $f(N)$ indicates that how many of the binary arrays of length $N$ don't consist of $3$ consecutive $1$'s. For example, if we'd have a look at $f(3)$: There ...
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### Determine convergence of series with natural logarithm [closed]

I am trying to determine the convergence of the series below: $$\sum_{n=1}^{\infty}{\frac{(n+1)}{(n^2+2)\ln(n+3)}}$$ I've tried comparison test, Cauchy-condesation, but nothing seems to work.
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### Mixing metric distance with Euclidean metric on $\mathbb{R}$

By far the most elegant proof of uniqueness of limits in a metric space I've seen (though I can't find the stack exchange link where it originated) is as follows. Let $X$ be a metric space, $(p_n)$ a ...
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### Do Individual Sequences Converge if their Sum Converges? [closed]

Is the following statement true? If the sequence $\{a_n + b_n\}$ converges and $\{a_n\}$ also converges then so does $\{b_n\}$
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### Proof: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$?

Question: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$? Answer: Thank to @TonyK @Ryszard Szwarc. I think that i found an ever stronger demonstration that ...
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### Let $0 < a_1 < a_2$. Consider the sequence $a_n$ defined by $a_{n+1}$ = $(a_n + a_{n−1})/2$. Show that lim $a_n$ = $(a_1 + 2a_2)/3$ [duplicate]

Let $0 < a_1 < a_2$. Consider the sequence $a_n$ defined by $a_{n+1} = \frac{a_n + a_{n−1}}{2}$. Show that $\lim _{n \to \infty}a_n = \frac{a_1 + 2a_2}{3}$. I have tried substituting multiple ...
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### Prove: in geometric sequence ($0\ <\ r\ <\ 1$) the ratio between a term and the sum of all following terms doesn't depend on the location of that term

Another question from my math finals, this time we were working with a geometric sequence. We were asked several questions about a specific sequence, but then the last question was this: prove that in ...
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### Prove a cauchy sequences [closed]

How to show that images problem is it true that the sequences convergen to x/y and that means the sequences is cauchy
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### How can I show a simple inequality? [duplicate]

I was looking for some paper, and I had a question about simple inequality process. How can we show that following inequality? $\sum_{t=1}^{T}\frac{1}{t}\leq 1+\log{T}$ For those who want a detailed ...
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### Is the following recursive sequence $a_n$ positive definite?

Is the following recursive sequence $a_n$ positive definite? $a_1=1\hspace{25pt}a_{n+1}= 2a_n +1$ I've been asked to prove also that's strictly increasing, both things by induction. Considering that ...
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### Coinciding Limsup and limit

Let $(x_n)_{n \geq 1}$ be a sequence of real numbers. Suppose that we are able to show that for a fixed number $m$, $(y_n)_{n \geq 1}:= (x_{n+m})$ and we know that $\lim_{n\to\infty}(y_n)=x$ for some ...
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### Proof verification that $\lim_{n \to \infty}\frac{n!}{n^n}=0$ (sequence)

Using an epsilon-N approach (since this is supposed to be a sequence), we require $$\forall \varepsilon>0, \exists N \hspace{1mm}\text{s.t} \hspace{2mm}n>N \implies |a_n-L|<\varepsilon$$ Now, ...
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### Harmonic Distribution of prime numbers [closed]

I developed a sieve that depicts the distribution of prime numbers as contained in harmonic (repetitive) patterns. Published it here What would be the process to know if I’m rightfully thinking this ...
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### How to bound $\sum e^{-a n^p}$

Let $0<p<1$ and $a>0$. Then it would seem that $$\sum_1^\infty e^{-an^p}\le Ce^{-a}$$ For some constant $C(p)$ since the terms in the summation decay exponentially. However, I can't quite ...
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### A combined arithmetic and geometric sequence question

Here is a question I am currently struggling with - The first, the tenth and the twentieth terms of an increasing arithmetic sequence are also consecutive terms in an increasing geometric sequence. ...
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