For questions about recurrence relations, convergence tests, and identifying sequences

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34 views

Is this series of products convergent/divergent

Please Help I've this series: $ \sum_{p=1}^q \prod_{k=p}^q \dfrac{n-k}{k}.\dfrac{1}{p(n-p)^2} $ with $1 \leq q \leq n/2$. Is this convergent or divergent? Many thanks!
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2answers
32 views

Proving measure of set is $0$

Let $\mu$ be measure on $(X,A)$ and $(A_k)$ sequence in $A$ such that $\sum_k\mu(A_k)<\infty$. Show that the set of points that belong to $(A_k)$ for infinitely many values of $k$ has measure ...
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1answer
29 views

Convergence of the limit of a sum of $|P_{nk}-P_{k}|$, where $P_{nk}$ and $P_{k}$ are sequences of nonnegative numbers summing to 1

Let $(P_{nk})_{k \geq 1}$, $n=1,2,\cdots$ and $(P_{k})_{k \geq 1}$ be a sequence of nonnegative numbers satisfying $\sum_{k=1}^{\infty}P_{nk}=1$ and $\sum_{k=1}^{\infty}P_{k}=1$, and let $\lim_{n ...
0
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3answers
161 views

Geometric sequence, finding two variables

In a geometric sequence $a=48, t_{n}= 16/27$ , and $S_n$= 976/27. Find n and r. This is what i have tried: $48, \ldots, 16/27$ One equation i got is: $16/27=48r^{n-1}$ I am not sure how to form ...
1
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2answers
101 views

Laurent Series Difficulty

Hello all at StackExchange, I'm having some trouble understanding computing the Laurent series for different domains. Here's my approach to finding the Laurent series for $\dfrac{3}{(z+1)(z-2)}$ for ...
0
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3answers
44 views

partial sum of geometrical sequence

Find partial sum of geometrical sequence with a2=12 and a5=324 Find $$\sum_{k=1}^\infty 4\left(\frac 1 2\right)^{k-1}.$$ Hope fully this makes sense. I am trying to understand how to do this. I ...
1
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1answer
26 views

$\|(I+A)^{-1}\| \leq \frac{1}{1-\|A\|)}$

I have the following problem, of which I have a slight problem to finish with the second part: Let $X$ be a Banach space and let $A \in B(X)$, $\|A\| < 1$. Prove that $(I+A)^{-1}$ exists and is ...
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3answers
328 views

First $3$ non-zero terms of the Maclaurin Series $\frac{1}{\sqrt{4+x^3}}$

Since each derivative will be multiplied by $3x^2$, are all the terms of this Maclaurin series $0$?
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1answer
46 views

simplify finite series 2 [closed]

I would appreciate if somebody could help me with the following problem Q: simplify infinite series $$\sin x+\sin 2x+\cdots+\sin nx(n\in\mathbb{N}, 0<x<2\pi)$$
2
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1answer
85 views

finding first 5 terms

Find first 5 terms of $a_n = 3(a_{n-1}+1)$ I believe this is fairly easy to do, I am unsure the proper format for this arithmetic sequence. Should it be 3(an-1)+1 with the +1 being the distance. Any ...
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2answers
100 views

Upper bound for $\sum_{j=0}^i {i \choose j}^{n}$

Is there an upper bound for sums of powers of binomial coefficients? I have $$\sum_{j=0}^i {i \choose j}^{n}$$ where $n$ is a positive integer. I am hoping this will help me solve Limit for a ...
2
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3answers
122 views

Show that : $ \sum^{\infty}_{n=1} \frac{ \sin(n \alpha)}{n} = \frac{\pi - \alpha}{2} $

Show that : $ \sum^{\infty}_{n=1} \frac{ \sin(n \alpha)}{n} = \frac{\pi - \alpha}{2} $ (Preferably) using complex analysis tools. Any hints or ideas is appreciated.
5
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2answers
104 views

Show that the sum of the series

Show that the sum of the series is greater than 24 $$\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\frac{1}{\sqrt{9}+\sqrt{11}} +\cdots+\frac{1}{\sqrt{9997}+\sqrt{9999}} > 24$$ I see ...
1
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3answers
45 views

if $S_{m}=\frac{m}{n},S_{n}=\frac{n}{m}(n\neq m)$,then $S_{m+n}$ and $4$ which is bigger?

let $\{a_{n}\}$ is arithmetic sequence,and $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$,and such $$S_{m}=\dfrac{m}{n},S_{n}=\dfrac{n}{m}(n\neq m)$$ then $S_{n+m}$ and $4$ which is bigger? My try: With out loss ...
13
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1answer
232 views

Prove $\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$

How can I prove that $$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$ I think this post can help me, but I'm not sure. Mathematica can provide a closed form for ...
2
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4answers
197 views

Every bounded monotone sequence converges

I am trying to prove: If a sequence is monotone and bounded then it converges. My idea is: Assume $a_n$ is monotone and not converges and then show that it is not bounded. But: my problem is ...
4
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1answer
31 views

Find asymptotics for solution $x$ of $(x+1)^{\frac{n+1}{n}}-x^{\frac{n+1}{n}}=5$

It is easy to see that for any $n\geq 1$, the equation $(x+1)^{\frac{n+1}{n}}-x^{\frac{n+1}{n}}=5$ has a unique positive solution ; call it $x_n$. Is there a simple asymptotic formula for $x_n$ ? I ...
2
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3answers
95 views

Finding a formula for a recursively defined sequence

I have a sequence given by: \begin{align} r_1 &= 1\\ r_2 &= 0\\ r_3 &= -1\\ r_n &= r_{n-1}r_{n-2} + r_{n-3}\\ R &= \{1, 0, -1, 1, -1, -2, 3, -7, -23, etc...\} \end{align} The ...
4
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2answers
62 views

An inequality with infinite sum

If $p>1$, and $k$ is a positive integer more than $1$, show that $$\sum_{n=2}^{\infty}\frac{(\ln n)^k}{n^p} \le \frac{k!}{(p-1)^{k-1}}$$ At first, I thought many ideas such as Cauchy-Schwarz ...
2
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1answer
56 views

Geometric Series, finding the r value

The sum of the first two terms of a geometric series is 4. The sum of the first four terms of the same series is 40. Determine the first five terms of the series. I am having trouble finding the r ...
7
votes
1answer
143 views

How to find closed form formula for a sum

I am a PhD student in electrical engineering. I need to find a closed form formula for the following series: $$\sum_{k=1}^{\infty}\frac{1}{2}A_k^2e^{-k^2\sigma_m^2}(e^{k^2\sigma_m^2}-1)$$where $A_k= ...
0
votes
1answer
74 views

Determine the explicit formula of the sequence.

Determine the explicit formula of the sequence: $ 4/5, 6/7, 8/9, 10/11,... $ I am unsure how to make the explicit formula for this. Is there a formula to follow, or do i just have to look for a ...
1
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2answers
48 views

Geometric Series

The first term in a geometric series is 4 and the sum of the first three terms is 64. Find the sum of the first eight terms of the series. I know the a value is 4, but I'm unsure of how to find the r ...
0
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2answers
60 views

(Geometric) Sum of $100-50-25-(25/2)+\ldots+ (25/16)$

Determine the sum for this geometric series: $100-50+25-(25/2)+\ldots+ (25/16)$ I found $7$ to be the number of terms in this series, and the sum of the series to be $67.1875.$, but, the ...
16
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4answers
522 views

Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$

I'm trying to calculate the limit for the sum of binomial coefficients: $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$
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1answer
22 views

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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0answers
77 views

DFT example in textbook

There is an example of the use of the DFT formula in my textbook which I don't quite follow. The text goes as follows: Let us define the $N$-periodic and anti-Hermitian series $g_n$ where $g_n = f_n ...
4
votes
1answer
73 views

A possible theorem

So i was playing around with members of a random power set, and i came to a revelation(at least to me it was :)). Say $A=\{1,2,3\}$ then for arbitrary $k,n\in Z^+$, $n=|A|$ and ...
5
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1answer
123 views

Closed form for $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2+a^2}$

I want to express $$\sum_{n=-\infty}^\infty \dfrac{1}{(z+n)^2+a^2}$$ in closed form. What comes to mind is the formula $$\pi\cot\pi z = \dfrac{1}{z}+\sum_{n\ne ...
2
votes
1answer
98 views

Prove $\liminf (s_n\cdot t_n) = \liminf (s_n) \cdot \liminf (t_n)$

I'm having problems with the following proof: If $s_n$ and $t_n$ are sequences, then does $\liminf (s_n\cdot t_n) = \liminf (s_n) \cdot \liminf (t_n)$? Is there a theorem that proves this? Is this ...
3
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1answer
40 views

Limit of product series for convergent and increasing sequences

Suppose $a_1,a_2,\ldots\in\mathbb{C}$ and $b_1,b_2,\ldots\in\mathbb{R}$. Suppose also that $\sum a_n$ converges, that $b_n\leq b_{n+1}$ for all $n\geq 1$, and that ...
4
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1answer
47 views

Where is sum $\sum_{n=0}^\infty \left(\frac{1-z}{1+z}\right)^n$ analytic

I'm trying to solve for the values of $z$ such that the function $$\sum_{n=0}^\infty \left(\dfrac{1-z}{1+z}\right)^n$$ converges, and also determine where the sum is analytic. Well, the series ...
2
votes
1answer
100 views

Is $\sum_{k=1}^\infty\frac{k^k}{k!}e^{-k}$ convergent?

I'm doing an exercise about the convergence of series: Is $$\sum_{k=1}^\infty\frac{k^k}{k!}e^{-k}$$ convergent? The following limit is given: $$ \lim_{k\to\infty}\frac{k!}{\sqrt{2\pi ...
0
votes
1answer
79 views

Find the partial sum of a given series?

I found this thread, but since it says that I can't ask for help there, I'm making a new one. How to find the partial sum of a given series? I have this series (WolframAlpha): $$ \sum_{n = 3}^\infty ...
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1answer
56 views

Series divergence - how precise should the answer be

Morning. I've written down some of my reasoning and arguments as to why the series diverges, however I am not certain I can safely conclude it diverges to $\infty$. Would you give it a look, please? ...
2
votes
3answers
408 views

Limit of the ratio of consecutive Fibonacci numbers [duplicate]

I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...
1
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1answer
57 views

Uniqueness proof of binary representation

I'm having trouble understanding this proof of uniqueness of binary representation of integers: Suppose there exists an integer $n$ with two different binary representations. Let these be: ...
1
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1answer
33 views

Question about the conditions that satisfy the Alternating Series Test.

The series $\sum_{n=1}^{\infty}(-1)^{n+1}u_n$ converges if all three of the following conditions are satisfied: The $u_n$'s are all positive. The positive $u_n$'s are (eventually) nonincreasing: ...
2
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2answers
243 views

Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent

$p>1$ is a integer, Show a convergent series $\sum\limits_{n=1}^\infty a_n$, $a_n\in\Bbb R$, such that the series $$\sum_{n=1}^\infty a_n^p$$ is divergent p.s. If $p>1$ is not an integer ...
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2answers
273 views

Real Analysis: Prove every unbounded sequence contains a divergent monotone subsequence

I have been struggling over this problem for hours, and I have no clue where to begin. Can anyone give a clear and complete proof of the following theorem. Every unbounded sequence contains a ...
2
votes
2answers
52 views

Upper bound for sum

I am trying to get an upper bound the following sum: $$S_{n,r}=\sum_{i=0}^n \binom{n}{i} \left(\frac{\binom{n}{i}}{2^n}\right)^{r} .$$ Any hints would be greatly appreciated. I thought of using ...
0
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2answers
42 views

To which value does $\lambda \sum_{j=0}^{\infty} \left[\left(1-\lambda \right)^j \right]$ converge to?

Suppose that $\lambda$ is smllaer than 1 and greater than 0. Then what does $\lambda \sum_{j=0}^{\infty} \left[\left(1-\lambda \right)^j \right]$ converge to? If this depends on specific range of ...
1
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1answer
53 views

Power series: If $|z-z_0|< R$ the series converges absolutely

I'm trying to prove absolute convergence of the power seris $$\sum_{n=0}^{\infty} a_n (z-z_0)^n, \qquad |z-z_0| < R$$ where $R^{-1} = \limsup |a_n|^{1/n}$. WLOG, suppose $z_0=0$ (otherwise ...
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1answer
63 views

Sequence of numbers with a special property [closed]

Prove that the sequence a(n) = 2013 + 317n, where n is any nonnegative integer, generates infinitely many palindromic numbers.
2
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1answer
84 views

Real Analysis: Unbounded Sequences

Currently, I'm working on the following problem about unbounded sequences and their subsequences. Though, I really don't understand how to prove the following, it appears to be a direct result of the ...
1
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2answers
90 views

question about taylor series

Can someone explain why 1 and 2 use different Taylor series? Why i cant use $1/(1+r)$ = $\sum_{n=0}^{inf}(-1)^n r^n$ on 2,vice versa?
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2answers
87 views

Calculate $\lim_{n \to \infty}U_n$

How can I calculate $\lim_{n \to \infty} U_n$ where $$U_n = \sum_{k=1}^n k \sin\left(\frac{1}{k} \right)?$$
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2answers
86 views

Repertoire Method. How to know when equations are valid

I'm trying to work a few examples from the Repertoire Method from the Book Concrete Mathematics. I'm working through the following recurrence: $f(0) = 1$ $f(n) = 2f(n-1) + n$ Which I generalized ...
1
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2answers
68 views

Show by induction $|a_1-a_2+a_3-\ldots \pm a_n| \leq |a_1|$

The assumptions are that $(a_n)$ is a decreasing sequence with $(a_n) \to 0 $, that is all terms are nonnegative. It is easy to see that the subtracted terms are always at least as great as the added ...
0
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0answers
31 views

Cesaro summability of a binomial series

For every $z\in\mathbb C$, put $$c_n(z)=\sum_{k=0}^{n-1}\mathrm{LCM}(1,2,\cdots,k)\binom{z}{k}$$ where $\binom{z}{k}=\frac1{n!}\prod_{j=0}^{k-1}(z-j)$ and LCM is the least common multiple. Is the ...