# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Limit of the sequence $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$.

I have tried to solve this limit : $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$. Where n $\in\mathbb{N}$. I have understood that the limit exists and goes to 0 if the argument becomes ...
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### Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
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### Finding limit of $\frac{a_n^3+5n}{a_n^2+n}$ for $(a_n)$ bounded.

Suppose that the sequence $(a_n)_{n \in \mathbb{N}}$ is bounded. Prove that the sequence $(c_n)_{n \in \mathbb{N}}$ defined by $$c_n = \frac{a_n^3+5n}{a_n^2+n}$$ is convergent and find its ...
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### To check whether series S and T are convergent or not [on hold]

To check whether series S and T are convergent or not . I applied ratio test for series S and found it to be convergent but i do not know about series T. Thanks
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### Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing?

Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing? I simplified it to $\frac{\sqrt{n^2+1}}{n}$, and I tried $a_{n+1}-a_n$ and $\frac{a_{n+1}}{a_n}$, but neither seem to work, how ...
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### Find the Summation of Summation [on hold]

An Array A consisting of N integers .We perform the following operation M times: for i = 2 to N: Ai = Ai + Ai-1 We have to find xth element of the array ...
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### Epsilon delta proofs of theorems of continuity

Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
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### Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
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### Is the sequence $s_n= \frac{\sin\frac\pi2}{1\cdot 2}+\frac{\sin\frac\pi{2^2}}{2\cdot 3} + \dots + \frac{\sin\frac\pi{2^n}}{n\cdot(n+1)}$ convergent? [on hold]

Which of following is correct? I think option D. Not sure though $$s_n= \frac{\sin\frac\pi2}{1\cdot 2}+\frac{\sin\frac\pi{2^2}}{2\cdot 3} + \dots + \frac{\sin\frac\pi{2^n}}{n\cdot(n+1)}$$ is ...
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### Harmonic progression sum

http://www.mathalino.com/reviewer/algebra/arithmetic-geometric-and-harmonic-progressions Please go to this link and see how they tell you to find the sum of harmonic prgression. However I am sure it ...
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### “Convergence”/“Divergence” of $\prod_{n=1}^\infty (1 - \gamma_n)$

While trying to understand a proof of a result in an article, I stumbled upon the product $$\prod_{n=1}^\infty (1 - \gamma_n)$$ with $\gamma_n$ a real scalar belonging to $(0,1)$. I'm not really a ...
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### Find the limit $\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)$

$$\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)\;=\;?\quad(n\in I) \\ \text{where \lfloor\cdot\rfloor is the greatest integer function.}$$ This is what I ...
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### At wich $n$ reaches the sequence its target value?

There are three parameters: $y_s=y[0]$ start value $y_t=y[n]$ target value $\alpha, 0>\alpha\leq1$ smoothness Starting at $y[0]=y_s$ the sequence is developed with this recursive formula: ...
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### Is it solvable to ask to find a recurrence relation for the following sequence 4,7,2,20,31,73,155,332,715,… ?

Is it solvable to ask to find a recurrence relation for the following sequence 4,7,2,20,31,73,155,332,715,... without any other information? If not, what would be the very least amount of information ...
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### How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?

We are given a fixed point on a circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with ...
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### Solve this puzzle? [on hold]

Given a number, the answer is a power of $2$. Given $1.000$ the answer is $16384$. Given $5.000$ the answer is $131072$. Can someone find a function, so given any number we can get the answer?
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