# 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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### Is the sum of partial sums $\frac{2\pi}{k}-{k \mod 2\pi}$ bounded?

Suppose, we are given sequence $a_k= k\mod 2\pi$ and sequence $y_{k,n}$, where $y_{k,n},0\leq n\leq k$ is just sequence $x_0,\dots,x_k$ sorted in increasing order for a given integer $k$. What can we ...
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### weakly convergent and the inner product in $l^{2}$

A sequence $\mathbf{x}^{(n)}$ in $l^{2}(\mathbb{R},\mathbb{N})$ converges weakly to $\mathbf{x}$ if, and only if, the sequence of inner products $(\mathbf{x}^{(n)} - \mathbf{x}, a)$ converges to $0$ ...
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### Evaluate: $\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$

The questions asks to evaluate: $$\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$$ The answer provided is $-\frac{1}{4}$, but I don't know how to solve it. I am not sure how to ...
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So here is the series: $\Sigma_{n=1}^{\infty} \frac{x^{2n}}{1+x^{4n}}$ $$\left| \frac{x^{2(n+1)} (1+x^{4n})}{x^{2n}(1+x^{4(n+1)})}\right| = \left|\frac{x(1+x^{4n})}{1+x^{4n+4}} \right| \overset{\text{... 1answer 25 views ### Given 2 numbers A and B how to check if (A+B) > (A*b)? [closed] Given 2 numbers  A  and B how to check if (A+B) > (A\cdot B), where + is concatenation ? e.g. A = 10, B = 2. So  A + B = 102 and A \cdot B = 20 so Here (A+B) > (A\cdot B) 1answer 35 views ### Is there a formula for finding the sum of square roots of first n natural numbers? [duplicate]$$\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5} ....... N,terms $$Is there a formula for finding the sum of square roots of first n natural numbers? 3answers 90 views ### Find \displaystyle\lim_{n\to \infty} \int_0^1 nx^n e^{x^2} dx The value of \displaystyle\lim_{n\to \infty} \int_0^1 nx^n e^{x^2} dx is ____________(round off to three decimal places) I tried integrating by parts and bring out some recurrence relation , but ... 2answers 52 views ### Evaluate \sum_{n=1}^{50}n\cdot n!. [duplicate] Evaluate \displaystyle\sum_{n=1}^{50}n \!\cdot\! n! I tried to write the sum like: 1+2\!\cdot\!2!+3\!\cdot\!3!+4\!\cdot\!4!+5\!\cdot\!5!+\ldots=1+4+18+96+600\ldots I can find a recursion like ... 1answer 31 views ### Consider the sequence u_n=\sum_{r=1}^{n}\frac{r}{2^r},n≥1. Find the limit of u_n as n\rightarrow\infty [duplicate] QUESTION: Consider the sequence$$u_n=\sum_{r=1}^{n}\frac{r}{2^r},n≥1$$Find the limit of u_n as n\rightarrow\infty MY APPROACH: I thought that this question has been asked before. But I am a bit ... 1answer 37 views ### Polynomial-Exponential Equation Consider the sequence$$ x_n = \sup \{ k \in \mathbb{N} : e^{ 2^{k}} 2^k \leq n \} $$I'm wondering if it possible to deduce that there exists \alpha \in (0,1) for which  e^{2^{x_n}} = O(n^... 3answers 58 views ### What is the sum of n terms 1.5.9+5.9.13+9.13.17…? How to solve this? My attempt is to write the r^{th} term which is$$(4r-3)(4r+1)(4r+5)$$Then let p=4r+1 The r^{th} term becomes$$(p-4)(p)(p+4)=p^3-16p$$Now we know summation \sum p^3 =[\... 0answers 16 views ### Density of tensor products of eventually zero sequences. Let c_{00}(\mathbb{Z}) denotes the space of eventually zero sequences over \mathbb{Z}. Let us define$$\mathcal{P}c_{00}(\mathbb{Z}^2)=span \{(a_k)_{k\in \mathbb{Z}^2 } : a_{(k_1,k_2)}=b_{k_1}c_{...
Supposing that $a_n$ is any sequence Knowing that$\displaystyle{\lim_{n \to \infty}n^4a_n=0}$, how can I show that $\sum_{n=1}^{\infty} a_n$ is convergent. Sorry but I really dont have any work to ...
### Comparing series $\ln(2)+\ln(\ln(2)+1)+\ln(\ln(\ln(2)+1)+1)+…$ to $\sum 1/k^2$
This is related to my previous question Does $\ln(1)+\ln(\ln(2)+1)+\ln(\ln(\ln(3)+1)+1)+...$ converge?. In the answers it is shown that $$\ln(2)+\ln(\ln(2)+1)+\ln(\ln(\ln(2)+1)+1)+...$$ diverges. In ...