# 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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### Find the value of question mark form the given information.

Find the value of question mark form the given
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### Space where normal convergence does not imply uniform convergence [closed]

We know that if we're in a Banach space, normal convergence implies uniform convergence. Is there an easy counterexample outside Banach spaces ?
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### Decide if the following series converge and explain your reasoning: $\sum_{n=1}^{∞}\frac{\cos(2n)}{n^3}, \sum_{n=1}^{∞}\frac{1}{4^n + 2n}$ [closed]

(a) $$\sum_{n=1}^{∞} \frac{\cos(2n)}{n^3}$$ (b) $$\sum_{n=1}^{∞} \frac{1}{4^n + 2n}$$ I'm a little confused on how to approach this problem -- any tips? Thank you!
1 vote
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### T\F: if $\lim_{k\to\infty}\sum_{i=n_k}^{n_{k+1}-1}|\alpha_i|= 0$ then $\sum \alpha_k$ converges.

Let $\sum \alpha_k$ be a bounded series of reals. Suppose $\{n_k\}_{k=1}^{\infty}$ is a strictly-increasing monotonic sequence of naturals s.t $lim_{k\to\infty} (n_{k+1}-n_{k})= \infty$. Prove or ...
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### Optimizing a recurrence relation for a sequence

Given the sequence $$a_k=\frac{(2k)!}{4^k(k!)^2(2k+1)}(0.5)^{2k+1},$$ I should find a recurrence relation for it. I came up with $a_0 = 0.5$ and $$a_{k+1}=\frac{(2k+1)^2}{8(k+1)(2k+3)}a_k.$$ is there ...
1 vote
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### Equivalence of expressions for potential of a line charge between two parallel grounded conductors - method of images and series solution

I am trying to figure out how two expressions for the potential of a line charge between two grounded parallel conductors are equivalent. Showing the equivalence of the two expressions seems more ...
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### How to find a closed-form expression for the sequence $x_n = x_{n-1}(1.1) + 100$

I have the sequence $\mathbf{x_n = x_{n-1}(1.1) + 100}$ with $\mathbf{x_0 = 100}$. How can I calculate $\mathbf{x_n}$ as an explicit function of $\mathbf{n}$?
1 vote
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### Why this sequence converges to the intersection?

Consider the following set up (you can follow on desmos): In a plane we have to circles that intersect at 2 point, circle $A$ and circle $B$, with centers at $A$ and $B$ respectively. For simplicity ...
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### Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$

Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$ I'm quite sure that this statement is true, but I'm having a little difficulty ...
1 vote
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### Confusing notation of sequences in $k$-dimensional Euclidean spaces.

Suppose $(x^{(n)})$ is a sequence in $\mathbb R^k$, $k \in \mathbb N$. From what I understand, $(x^{(n)})$ is a sequence of sequences $(x_1^{(n)}, x_2^{(n)}, x_3^{(n)}, ..., x_k^{(n)})$. Or in other ...
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### Trouble understanding difference between the following two results involving the number of terms in the set $\{n: s_n \gt \lim \sup s_n\}$

There are two results written in Ross' book on Elementary Analysis, which state the following - If $L = \limsup s_n \neq \infty$, then for every $\alpha > L$, the set $\{n: s_n \gt \alpha\}$ is ...
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### How to study the convergence of a piecewise serie?

I try to study the convergence of the series of the general term: $$U_n=\begin{cases} \frac1n ,\text{ if n is square}\\ \frac1{n^2},\text{ else } \end{cases}$$ But I didn't find a way to start, even ...
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### Interesting infinite product $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$

I have found an interesting family of infinite products. The most interesting one of them being: $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$...
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### Sequence or probability problem [closed]

What way should be continued the sequence 1 2 3 4 5 6 7 8 9 10? My answer is the sequence above representss natural numbers sequence. The sequence is bounded below (1) and does not have upper limit, ...
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### Distribution of $Y = X \bmod 2\pi$ with $X$ being a Cauchy distribution

Let $X$ be a Cauchy distribution with parameter $\theta$, that is to say, its density function is: $$f(x;\theta) = \frac{\theta}{\pi(x^2 + \theta^2)}$$ I'm asked to get the distribution of $Y$ where ...
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### Is this sequence arithmetico geometric? [closed]

I am im desperate need of help to solve a mathematic exercise. I have to determine if this sequence is arithmetico geometric but I struggle explaining it.. I am supposed to present it in front of my ...
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### Show $\epsilon[\frac{x}{\epsilon}] \rightarrow x$ when $\epsilon \rightarrow 0$

The question is to show that $\epsilon[\frac{x}{\epsilon}] \rightarrow x$ for $\epsilon \rightarrow 0$. Here, $x \in \mathbb{R}^n$ and $\epsilon \in \mathbb{R}$. Intuitively, it does not make sense to ...
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### Problem in proof of theorem involving $\lim \sup$ and $\lim \inf$

Source: Elementary Analysis By Kenneth A. Ross Theorem: If $s_n$ converges to a positive real number $s$ and $(t_n)$ is any sequence, then $$\lim \sup s_nt_n = s\cdot\lim \sup t_n$$ Proof: We first ...
### Find least $\lambda$ for which recursive sequence is always positive
Find the least $\lambda$ for which the sequence $\{b_n\}$ defined by $b_1=1$, $b_2=\lambda-1$ and $b_{n+2}=\lambda(b_{n+1}-b_n)$ is always positive. I guess $\lambda=4$, which yields \$b_n=2^{n-1}(n+1)...