# 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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### Better understanding of a series

Let $B=\{x\in\mathbb{R}^N:\ |x|<1\}$ with $N\geq 2$ and $x_n$ a countable dense set in $B$. Consider the function $$u(x)=\sum_{i=1}^\infty\frac{1}{2^{i}|x-x_i|^{1/2}},\ \forall\ x\in B$$ By using, ...
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### Positive limit of sequence vs. positive terms

Let $\{x_m\}$ be a sequence in $E_1$ that converges to $L \in E_1$. a. prove that if $L>0$ and there exists $n \in N$ such that for all $m >n$ holds that $x_m > 0$ b. True or false? If for ...
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### Show a series of functions has a continuous sum.

For every $x \in \mathbb{R}$ define $$I(x) = \begin{cases} 0 & \text{if} & x \leq 0,\\ 1 & \text{if} & x > 0 \end{cases}$$ Suppose that $(x_n)$ is a sequence of points in $(a, b)$ ...
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### Convergence of (infinite) potence tower?

Let $(a_n)_{n\in\mathbb{N}}$ be a series in $\mathbb{C}$ or $\mathbb{R}$. Which contraints must $(a_n)$ match to make $b_n := a_1^{a_2^{...^{a_n}}}$ converge for $n\rightarrow\infty$? For constant ...
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### Cauchy Sequences bounded away from zero.

Is there a positive real number which can be written as a Cauchy sequence, such that this Cauchy sequence is bounded away from zero and also this sequence contains infinite number of positive and ...
I am stuck in my one of the homework problems, the question is like the following: Let $(x_n)$ be a bounded sequence, and let $c$ be the greatest cluster point of $(x_n)$: (a) Prove that for every $... 0answers 3k views ### Algorithm for conversion of seconds into date/time? What mathematical algorithm would I use to convert seconds since 1/1/1970 into a usable date/time format (time can be left off)? This includes leap years with extra days and any other additions. I'm ... 1answer 41 views ### Is there a specific terminology for numbers which are nontrivial multiples of triangular numbers? (Note: Please see this new question for the motivation.) A number$T$is said to be triangular if it could be written in the form $$T=\frac{n(n+1)}{2},$$ where$n$is a positive integer. Here is my ... 1answer 142 views ### Find a formula of sequence:$\frac{-1}{2}, 0, \frac{1}{10}, 0, \frac{-1}{26}, 0, \frac{1}{50}, 0, \frac{-1}{82}, 0, \frac{1}{122}, 0, \dots$I'm working on a discrete math homework that finding a formula for the following sequence: $$\frac{-1}{2}, 0, \frac{1}{10}, 0, \frac{-1}{26}, 0, \frac{1}{50}, 0, \frac{-1}{82}, 0, \frac{1}{122}, 0, \... 2answers 58 views ### Uniqueness of coefficients in infinite sin series a(x) = \sum\limits_{n=1}^{\infty} a_{i} \sin(n x) =0 if we can say$$a(x) = \sum\limits_{n=1}^{\infty} a_{n} \sin(n x) =0, \forall x$$does this generally imply that all constant coefficients a_i should be zero, or can I construct any number of ... 3answers 92 views ### Solving a recurrence relation: can't figure out how to convert from summation I am really struggling to solve this recurrence.$$ T(n) = T(\sqrt{n}) + n. $$I am asked to give asymptotic upper and lower bounds for T(n). I am free to use any method to arrive at my answer, ... 1answer 55 views ### Find the recurrence formula! I have a sequence defined by recursion as follows:$$\begin{cases}x_0=a\\ x_{n+1}=x_n\cdot B^{x_n} \end{cases}$$where a,B are fix natural numbers. Does anyone know how to find a recurrence formula ... 1answer 53 views ### Finding the common difference and hence, the sum of an A.P Find the sum to 25 terms of an A.P with the first four terms as 1, \log_yx, \log_zy,-15\log_x z. My attempt: I started out with, 2\log_yx = 1+\log_zy and, 2\log_zy = \log_yx -15\log_xz ... 1answer 61 views ### Uniform convergence of U_n(x) = \sum_{n=0}^{+ \infty} (-1)^n \ln ( 1 + \frac{x}{1+ nx} ) . We consider the series of functions:$$U_n(x) = \sum_{k=0}^{n} (-1)^k \ln \left( 1 + \frac{x}{1+ kx} \right) ,~ x \geq0.$$Prove that U_n is convergent. Study the uniform ... 1answer 69 views ### Harmonic series (Maths and Music) I am a high school student trying to apply calculus to music (harmonic series). I am just wondering, how can I collect data from any online music app (with music tones that form harmonic series - ... 1answer 37 views ### Prove that the function can be continued into a larger domain Prove that the function f(z)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{z^n}{n} can be continued into a larger domain by means of the series$$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\... 1answer 61 views ### A conjecture on bounded complex partial sums A friend of mine has made the following conjecture, but we don't know how to prove it. Let$(a_n)_{n\in \mathbb {N}}$be a strictly increasing sequence of natural numbers. Suppose that for every ... 1answer 139 views ### When is the product of limsups equal to the limsup of the products? Let$\left\{x_{n}\right\}$be a sequence of real numbers where$x_{n} > 0$for all$n \in \mathbb{N}$. Given that \begin{equation*} x = \limsup_{n \rightarrow \infty} x_{n}^{1/n} = \limsup_{n \... 2answers 41 views ### Technique for using direct and limit comparison tests? Suppose you have an infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^2 + 9}$$ Using the direct comparison test, the sequence can obviously be compared to$\sum_{n=1}^\infty\frac{1}{n^2}$since it is ... 1answer 66 views ### For which value of$x \in \mathbb{R} $does$\sum\limits_{n=1}^\infty x^{n\log n}$converge? For which value of$x \in \mathbb{R} $does the following series converge: $$\sum_{n=1}^\infty x^ {n\log n}.$$ The series of the absolute values is $$\sum_{n=0}^\infty |x|^ {n\log n},$$ and ... 1answer 79 views ### How to solve the system$x_{t+1}=-x_t-2y_t+3t-2$,$y_{t+1}=-2x_t+2y_t+t+1\$
I have the following system of recurrence equations: $$x_{t+1}=-x_t-2y_t+3t-2\qquad y_{t+1}=-2x_t+2y_t+t+1$$ I write this in matrix-vector form: $$r_{t}=Ar_{t-1}+b\cdot (t-1)+c$$ I repeatedly ...