# 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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### How can one show that $(U_n)$ is arithmetic progression

Given that $(U_n)$ a numerical sequence such that : $U_0$=3 $U_{n+1}=\sqrt{(U_n)^2+8n+16}$ Show that $(U_n)$ is a arithmetic progression. So I have to show that $U_{n+1}-U_n=r$ where $r$ is a ...
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### proof of divergence of the serises

Let $\sum c_n$ be a series of positive numbers. Assume that $$\lim_n {{c_{n+1}}\over {c_n}} = r$$ If $0\lt r\lt 1$ then the series converges ; if $r\gt 1$ then diverges . Now , if $r\gt 1$ ...
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### Convergent subsequences common to two bounded sequences

Suppose $( a_n ) _n$ and $( b_n ) _n$ are two sequences of real numbers (not necessarily Cauchy or convergent) Suppose $| a_n | < 2 \ \forall n$ and $| b_n | < 17 \ \forall n$. Prove that there ...
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### Limit is infinite or finite?

The given function is $${{Log\ z}\over {z-1}}=1- {1\over 2}(z-1)+ {1\over 3} (z-1)^2-{1\over 4}(z-1)^3+....$$ Then it is said that the function tends to $+\infty$ as $z$ tends to $0$ . But ...
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### How to prove that these sequences both converge and have the same limit

Prove that the sequences $\frac{2a_nb_n}{a_n+b_n}$ and $\sqrt{a_nb_n}$ are both convergent to the same limit for all ai,bi>0. Given the special case of two positive numbers $a_i$ and $b_i$, the ...
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### Question about the Convergence of Infinite series due to Cauchy

If there exists $r \in \mathbb{R}$ with $r < 1, K \in \mathbb{N}$, such that $|X_n|^{1/n} < r$ for all $n > K$, then the series $X_n$ (Summation of $X_n$ from $n$ to $\infty$) is absolutely ...
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### How to find value of i when ∑ from k=1 to i is defined by a recursive formula and equals 982?

Thanks for the pointers! Here's updated and edited question I'm trying to find the number of days it takes to reach 982 miles when you start traveling at 18 miles/day and decrease your speed by 2% ...
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### Center of gravity of a sequence

I have a problem to solve that consists in finding a frequency domain expression of this expression, the center of gravity of a sequence. I have tried in several manners but no sucess so far. Does ...
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### Epsilon-N limit Proof for product law for limit

Let the $\lim_{n\to \infty} x_n=a$ and let $\lim_{n\to \infty} y_n=b$. Let $\epsilon>0$ for some $n\ge N$ My notes says: $|(x_n)(y_n)-ab|=|x_n(y_n-b)+b(x_n-a)|$ Can someone show me the ...
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### Decide convergence of the series .

I have problem with these two: a) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{(\ln{\ln{n}})^{\ln{n}}}$ b) $\displaystyle \sum_{n=3}^{\infty} \frac{1}{n \cdot \ln{n} \cdot \ln{\ln{n}}}$ My try: a) ...
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### Evaluating limit of a sequence

Prove: $$\lim_{x\, \to \,-\infty}⁡ \dfrac{ x^2+2x-3 }{ x^2+1 }=1$$ While making evaluations on my draft, i get: $|f(x)-L|= \dfrac{2}{|x-1|}$ I want to "remove" the absolut value in order to find ...
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### Infinite series convergence question

$$\sum_{n=3}^{\infty}\frac{(-1)^n}{\log n}$$ Can the conditional convergence of this series be proved by alternating series test, since you need n to be a natural number for the alternating series ...
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### convergence/divergence problem

Does the following series absolutely converge, conditionally converge or diverge? $$\sum_{n=1}^{+\infty}\sin(n^2)\sin\left(\frac{1}{n^2}\right)$$ I don't even know where to begin, I tried the limit ...
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### Question on the proof about simple harmonic series..

This is humble proof about harmonic series on my own. 1 + 1/2 + 1/3 + 1/4 + 1/5....... = 1 + (1 - 1/2) + {(1 - 1/2) - (1/2 - 1/3)} + {(1 - 1/2) - (1/2 - 1/3) - (1/3 - 1/4)} +... = 1 + (1/2)n - (1/6)...
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### Prove that $\frac{1}{\sqrt{1-\sin^2{x}}}=\sum\limits_{n=0}^{\infty} \frac{(2n)!(\sqrt{\sin{x}})^{4n}}{4^n (n!)^2}$.

Prove that $$\frac{1}{\sqrt{1-\sin^2{x}}}=\sum_{n=0}^{\infty} \frac{(2n)!(\sqrt{\sin{x}})^{4n}}{4^n (n!)^2}$$
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### Showing the limit exist by using sequences

$$a_1=1,a_{n+1}=\sqrt{6+a_n}, n \in \mathbb{N}$$ Show that the limit $\lim\limits_{n \to +\infty} a_n$ exists and find it. I know to prove that the sequence is bounded and monotonous but I still ...
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### Can this sequence be expressed with a closed-form formula?

1, 3, 5, 7, 9, 13, 17, 21, 25, 31, 37, 43, 49… So if you don't count the first number the sequence is +2 for four times. After that it changes to +4 for four times, then +6, +8 and so on, all for ...
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### How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

Definitions: Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has ...
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### Finding sum of two terms in a geometric progression

If $36, p, \dfrac 94, \mathrm{and}\,q$ are consecutive terms of a sequence, what is the sum of $p$ and $q$. I don't even know where to start.
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### What is the easiest approch to solve this sequences and series problem?

I've attempted this problem by counting pages, which is a tedious approach, is there a shorter method?
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### Use Taylor Theorem to find the polynomial approximation

A) Find a polynomial approximation for $f(x)=2e^x$ centered at $0$ for values of $x$ in the interval $[-1,1]$ B) what is the actual bound on the error in your approximation given by Taylor theorem? ...
Please I have tried to show that 1) $\sum_{j=1}^{N-1}\cos^2(\frac{2\pi}{N}\cdot j\cdot\Delta c)=N-1$ for $\Delta c=\frac{N}{2}$ and that 2)\$\sum_{j=1}^{N-1}\cos^2(\frac{2\pi}{N}\cdot j\cdot\...