# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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### What was Euler's misconception about functions and infinite series?

I just read this on Strichartz' The way of Analysis: [...] Euler - the leading mathematician of the eighteenth - developed all techniques needed for the study of Fourier series, but he never ...
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### Frequency analysis/discrete uniform distribution in multiple choice tests

I may be using the wrong terms in the title but I read that if something is random then each character will occur an equal amounts of times. I read this when reading about the One-Time Pad cipher, ...
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### representing a recursive difference equation of two variables into one variable equation

suppose the following recursive difference equation ($t$ is time): $$x_t = \frac{a}{1+a}x_{t-1} + \frac{1}{1+a}x_{t+1}$$ where $0<a<1$ is assumed and all values of $a$ at past times are ...
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### Validating a proposition

Proposition: For all $k,n\in\mathbb{Z^+}$ $s.t$ $n\lt4$ $2{n\choose n}+{n\choose n-1}+...+{n\choose k-(n-2)}=2^n$ for $1\le k\le n-1.$ I understand that this proposition is invalid, so are there ...
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### Sequence with a fixed last element Notation

I was trying to write a sequence of two different elements (that always appear in order) with a fixed last element, for an example: $A_1, B_1, A_2, B_2, A_3, B_3, A_4$. I'm not sure which would be the ...
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### Hypothesis testing for equivalence of two arrangements

I have two arrangements(i.e. permutations) of numbers. First one is the target/real arrangement. Second, is the observed arrangement. e.g. Target := 1,2,3,4,5,6,7 Observed := 4,1,7,3,2,5,...
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### Is something wrong in this proof?

Show that if $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n| > 0$, then $R\leq 1$. Proof: Suppose that $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n|=\alpha > 0$. ...
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### Proving that $\sum_j x^j$ is differentiable $(-1,1)$

I'm really struggling with understanding how to apply the Weierstrass M-test, and so some hints on this question would be much appreciated: First I want to prove that $\sum_{j = 0}^\infty x^j$ is ...
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Let $(a_n)_{n \in\ \mathbb{N}}$ a bounded sequence in $\mathbb{R}$. For $n \in \mathbb{N}$ let $$v_n=\sup\{a_k; ~k \geq n\},\quad u_n=\inf\{a_k; ~k \geq n\},\quad s_n=\sup\{|a_k-a_l|; ~k,l \geq n\}... 1answer 102 views ### examine the convergence of the series$$a_n=\frac{1}{n*10^{log(logn)}}$$I have to examine \sum{a_n} so I used Cauchy's condensation test and I got:$$b_n=\frac{2^n}{2^n*10^{log(log2^n)}}$$so$$b_n=\frac{2^n}{2^n*10^{log(log2)^n}}$$... 1answer 21 views ### Radius of convergence of a series (text problem) the series: \sum_{n=1}^\infty a_nx^n where (a_n)_n is a limited sequence with L((a_n)_n) \subseteq \mathbb{R}\backslash \{0\} My main problem is to get to something to work with. I dont know ... 0answers 70 views ### Recurrence in two variables Anyone know how to solve the following recurrence relation in two variables:$$ f(x,y) = b f(x-1,y) + c f(y,x-1), \qquad \begin{cases}f(x,0) = b^{(x-1)} \\ f(0,y) = 0 \end{cases} $$(Note: repost of ... 1answer 71 views ### 'ϵ-δ' proof for the following sequence I need help writing a formal 'ϵ-δ' proof for the following sequence:$$ \lim_{n\to \infty}(n+2)^2 \sin(1/n)=\infty $$Thanks in advance. 1answer 54 views ### Check definition-Decreasing sequence Is the following math definition of a decreasing sequence from a certain range correct? \exists n_0. !n. n \ge n_0 \Rightarrow f(n+1) <= f(n) I mean by "from a certain range", that when n \ge ... 1answer 33 views ### Error estimation help I'm supposed to find a Taylor polynomal of the n^{\text{th}} degree, where x = a, and estimate the error for the given interval. The problem I'm given is:$$f(x) = \sqrt{x}, a = 4, n = 2, 4 \leq x ...
consider $\displaystyle \sum_{n=1}^\infty (-1)^{n-1}a_n$ where $(a_n)$ is a monotone decreasing sequence of nonnegative numbers with $a_n \rightarrow 0$ by the alternating series test, series ...