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

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1answer
23 views

Limit of quotients implies limit of roots

If $a: \Bbb N\to (0, \infty)$ is a sequence with $\lim a_{n+1} / a_n = L < \infty$, then it has to be shown that $\lim a_n)^{1/n}$ = L. Has anybody a hint for me? I tried using the definition ...
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0answers
114 views

Combinatorics : number of non-decreasing series of r distinct numbers where the size of series ranges from 1 to N

I understand that number of non decreasing sequences of size M with N distinct numbers is (N+M−1)C(M). However, I'm interested in finding out the number of such series of r distinct integers where the ...
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0answers
43 views

What is the sum of reciprocals of Natural Numbers? [duplicate]

I want to calculate the sum of first $n$ natural numbers. I used the following C program to compute the first '$n$' digits : ...
3
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4answers
76 views

if $|a|<1$ so $\lim_{n\to \infty}na^n=0$.

Prove that if $|a|<1$ ($a$ is real) so $\lim_{n\to \infty}na^n=0$. I know that I need to use squeeze theory (because I have $-1<a<1$) but I dont see how. thanks
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1answer
78 views

Why $\ln n<n^{1/4}$?

In Calculus Book of "THOMAS" page 561 Twelfth Edition, I found this example Example 3 $\quad$ Does $\displaystyle \sum_{n=1}^\infty \frac{\ln n}{n^{3/2}}$ converge? Solution $\quad$ Because $\ln n$ ...
-1
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1answer
41 views

Series Summation,Convergence

I am stuck on the 4 th one.I have done the rest.I have found out the value of a_n.But not getting how to proceed for the 4 th one.
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1answer
15 views

Proof that for two converging sequences and two corresponding unbound sets - the limit is equal.

Ok so I know I've seen this before phrased differently but I can't quite put my finger on the solution. The question goes as follows: Let $a_n, b_n$ be two converging sequences. Also, let ...
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1answer
27 views

Summation over combinations

Can I reduce the following sequence into a single term. If so someone please point me in the right direction? $$ \Gamma = \sum_{i=1}^{r} \binom{n-i}{r-i} \, $$
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1answer
37 views

The equality $\frac{3}{4}\zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} - \sum_{n = 1}^\infty \frac{1}{(2n)^2} = \sum_{n = 0}^\infty \frac{1}{(2n + 1)^2}$

I was reading an article and the author immediately states that the following identity is clear $$\frac{3}{4}\zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} - \sum_{n = 1}^\infty \frac{1}{(2n)^2} = ...
6
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1answer
104 views

Let $n$ be a positive integer. Show that the smallest integer greater than $(\sqrt{3} + 1)^{2n}$ is divisible by $2^{n+1}.$ [duplicate]

Let $n$ be a positive integer. Show that the smallest integer greater than $(\sqrt{3} + 1)^{2n}$ is divisible by $2^{n+1}.$ We know that $(\sqrt 3-1)^{2n}<1$
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1answer
46 views

Sum of Arithmetic series

Sum of consecutive values can be found easily. But I can't figure it out how to find the closed form of the following arithmetic series? Can anybody explain it elaborately? $ S = (1) + (1+2) + ...
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3answers
89 views

Can someone explain how my teacher found a formula for the nth partial sum of a series?

I'm trying to understand how he did this. Can someone explain this to me step by step? Did he combine $(-1)^{n-1} \cdot \dfrac{1}{2^{n-1}}$ together?
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1answer
23 views

Arithmetic and Geom. sequences

Find the arithmetic sequence a, b, c, d if (a-2), (b-4), (c-3), (d+2) is a geometric sequence. I've tried setting up ratios with the terms of the geometric sequence, but I'm not sure how to solve ...
3
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1answer
139 views

Find the value of $\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+ … }}}}$

I have tried solving this question. It would be great if someone gives some idea about how to go about solving this question.
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3answers
88 views

does the series $\sum_{n=1}^{\infty }\frac{\sin(n)n!}{n^n}$ converge?

I get to the point, using the ratio test: $$\lim_{n \to \infty }\frac{\sin(n+1)n^n}{(n+1)^n\sin(n)}$$ and I know I can take a $1/e$ out but I'm not sure what to do from there.
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1answer
40 views

limit of a geometric mean

Suppose that a sequence $a_n$ of positive numbers converges to $a$. Show that $$\lim_{n\rightarrow \infty}\left(\prod_{i=1}^{n}a_i\right)^{1/n}=a$$ This seems to be simple using that $x=e^{\log x}$, ...
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4answers
60 views

Does $n^n = n! = (n+1)^n$ when computing a limit as n approaches infinity

I am performing the ratio test on series and I come across many situations where I have any of those 3 in the numerator and denominator and I was wondering what I could cancel as n approaches ...
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1answer
39 views

Prove $a_n = 7a_{n-2} + 6a_{n-3}$ for $n\ge 3$

Let $a_0 = 1$, $a_1 = 1$, and $a_k = 2a_{k-1} + 3a_{k-2}$ for all integers $k\ge 2$. Prove $a_n = 7a_{n-2} + 6a_{n-3}$ for all $n\ge 3$. Proof: Let $n\ge 3$ be arbitrary and fixed. From here, I ...
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4answers
60 views

Limit of a sequence - the sign of infinity

I need to evaluate the following limit: $$\lim_{n\to\infty}\frac{n^7-2n^4-1}{n^4-3n^6+7}$$ which isn't very hard. I divided the numerator and the denominator by $n^7$. The problem is with the sign ...
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1answer
40 views

Unordered convergent sums

In Elementary Real Analysis of Thomson and Bruckner, I'm stuck in exercise 3.3.4 page 88. Could you please help me giving me hints? Let the infinite index set $I$. Show that if $\sum\limits_{i\in ...
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2answers
127 views

If $\lim_\limits{n\to\infty}(a_n\sum_{i=1}^{n}a_i^2)=1$, then $\lim_\limits{n\to\infty}\sqrt[3]{(3n)}a_n=1$

If $\lim\limits_{n\to\infty}(a_n\sum\limits_{i=1}^{n}a_i^2)=1$, prove $\lim\limits_{n\to\infty}\sqrt[3]{(3n)}a_n=1$. Since $a_n^2\geqslant0$, if$\sum\limits_{i=1}^{n}a_i^2\to M,a_n\to 0, $then ...
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2answers
60 views

Infinite Binomial coefficient sum

$$\sum_{k=1}^{\infty} \binom{n}{k}$$ I am trying to do this, I think $|n| < 1$ is implied but Im not sure? $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ $$S = n! \cdot \sum_{k=1}^{\infty} ...
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2answers
72 views

Find formula for nth term of series (confused how to do it).

So my question is find a formula for the sum of the first $n$ terms of the series (not sequence): $$\frac{4}{81}+\frac{4}{729}+\frac{4}{6561}+\frac{4}{59049}+\ldots~\textrm{upto }n\textrm{ terms}$$ ...
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2answers
67 views

$\sum a_n$ converges conditionally

If we assume that $\sum a_n$ converges conditionally then How can we comment that $\sum a_{2n} $ does not converges, While it does when $\sum a_n$ converges absolutely ?
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2answers
71 views

calculate the intersection of two number series

I have a series of numbers. It is in the form of a parabola. This series is guaranteed to have at least one perfect square within it (edited I thought there was only one). The second series is also a ...
4
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0answers
42 views

Determine all sequence $a_{n}$ such $ a_{n+2}$ is divisible by $a_{n}$

Determine all sequence $a_{0},a_{1},a_{2},\cdots$ of postive integers with $a_{0}\ge 2015$ such that for all integers $n\ge 1$ (1)$ a_{n+2}$ is divisible by $a_{n}$ (2) $|S_{n+1}-(n+1)a_{n}|=1$ ...
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0answers
50 views

Frequency of non-increasing and non-decreasing subsequences in Matlab

Having a sequence of numbers of length L, I need to count how many non-decreasing and non-increasing sub-sequences of exact length are there. For example, if I have a sequence of length 15 $$2, 4, ...
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3answers
121 views

$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$

What tools would you gladly recommend me for computing precisely the limit below? Maybe a starting point? $$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$$
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1answer
28 views

Relation between LCM of terms of sequence with sum of sequence

Is there any relation between LCM of some arbitrary sequence and sum of elements of sequence ? How to find the LCM if only sequence sum is given in short time ?
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2answers
72 views

Evaluating $\sum_{n=1}^{\infty}\frac{n}{(n+2)!}$

I need to evaluate $$\sum_{n=1}^{\infty}\frac{n}{(n+2)!}$$Answer in book and WolframAlpha both say that is equal $3-e$. Thus, I have mistake and got: $$ \begin{align} ...
2
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1answer
113 views

Prove that $1-\frac{1}{1+\frac{\alpha}{nm}}\leq \sqrt{1-\frac{1}{1+\frac{\alpha}{n}}}\sqrt{1-\frac{1}{1+\frac{\alpha}{m}}}$.

For what values of the real parameter $\alpha$ the following inequality is true? $$1-\frac{1}{1+\frac{\alpha}{nm}}\leq \sqrt{1-\frac{1}{1+\frac{\alpha}{n}}}\sqrt{1-\frac{1}{1+\frac{\alpha}{m}}}$$ for ...
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0answers
65 views

trying to prove that a sequence to power of 2 converges too

If: $$\lim_{n\to \infty}a_n=L$$ So: $$\lim_{n\to \infty}a_n^2=L^2$$ ..... my try to prove: Let $\varepsilon>0$. because $$\lim_{n\to \infty}a_n=L$$ so $$\forall\varepsilon>0,\exists N: ...
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2answers
70 views

Given a series how do I create a function to describe that series

Given a series of numbers how do I find the equation that describes the series? For example given the following series of numbers... $$352, 1424, 2528, 3664, 4832, 6032, 7264, 8528, 9824, 11152, ...
1
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1answer
71 views

Prove sequence converges to zero

There are two sequences $(a_n)$ and $(b_n)$ and I know that their multiplication $(a_n b_n)$ converges to 0. Let there be a constant $c>0$ that for almost every $n$, $b_n\geq c$. I need to prove ...
3
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1answer
52 views

Why can you place in the recursive definition to find the limit?

When required to find limits of recursive sequences, i.e. $$x_{n+1}=\frac{1}{4-x_n}\qquad x_0=3$$ The steps are usually pretty consistent. First you prove it's monotonous and bounded, and therefore ...
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3answers
92 views

How to find the Maclaurin series for $e^{-x^2}$

I don't know how to get $$1-x^2+\frac{x^4}{2!}-\cdots.$$ I think it is too complex, if not impossible, to just use the definition of Maclaurin series. Using the definition: consider the situation ...
0
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1answer
28 views

Explain the sum of this series

$\Sigma_{n=0}^{\infty} i^n = \frac{1}{1-i}$ How is this the case? I can't find any reference to this anywhere but I know it can be used to solve the question: $\Sigma_{n=1}^{\infty} ...
0
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1answer
145 views

Is there a closed-form of $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^4}$

Is there a closed-form summation result for Fourier series: $$\sum_{n=1}^{\infty}\frac{\sin(n)}{n^4}?\tag{1}$$ I tried using available result of the following (odd) function : ...
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2answers
54 views

Divergent sequences whose sum diverges?

Are there any divergent sequences whose sum also diverges? Can anyone please explain with an example? Thank you.
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1answer
33 views

Study the convergence of $\sum_n\left|\frac{\sin(\alpha_n-\alpha_m)}{\alpha_n-\alpha_m}\right|$.

In many applications available on Math Stack Exchange: When does $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$ absolutely converge? Does $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converge conditionally? ...
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2answers
88 views

Proving $\frac{1}{\sqrt{2}}=1-\frac{1}{2^2}-\frac{1}{2!2^4}-\frac{3!!}{3!2^6}-\frac{5!!}{4!2^8}-\cdots$

How can I prove $$\frac{1}{\sqrt{2}}=1-\frac{1}{2^2}-\frac{1}{2!2^4}-\frac{3!!}{3!2^6}-\frac{5!!}{4!2^8}-\cdots$$ I wanted to prove it by using the Taylor series of $\sqrt{2}$, but I couldnt do.
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2answers
58 views

Interpretation of Zeno's paradox in Gelfand's Algebra text.

Imagine that achilles is running ten times slower than the turtle. The resulting geometric series will be: $ 1+10+100+\dots$ whose sum is computed to be $-\frac 19$. Previously in the text the ...
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1answer
33 views

$\sum^n_{r=0} (-1)^r C_r(1/2^{r}+3^r/2^{2r}+7^r/2^{3r}+\cdots\infty)$

$$\sum^n_{r=0} (-1)^r C_r(1/2^{r}+3^r/2^{2r}+7^r/2^{3r}+\cdots\infty)$$ is equal to? How to approach this problem?Hints please!!! BTW $C(r)$ stands for $(n)C(r)$
1
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1answer
32 views

How to show $n \sum_{k>n} (k^2 \log k)^{-1} \sim (\log n)^{-1}$?

How does one show that $$n \sum_{k>n} \frac{1}{k^2 \log k} \sim \frac{1}{\log n} \quad ?$$ Many thanks for your help.
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0answers
29 views

Seeking a closed form for the derivatives of a divergent series of functions

To begin with I have a series of interest, $\sum_n f_n$ that diverges everywhere $f_n: \mathbb{R}^2 \rightarrow \mathbb{R}$. However, $\int_z \sum_n \partial_z f_n$ has a nice closed form, lets call ...
0
votes
1answer
46 views

Number of stops an elevator will make

For $Y \sim \text{Poisson}(10)$, assume $Y$ people get into an elevator on the ground floor. There are $n$ floors above the ground floor, and everyone is equally likely to get out on any of them, and ...
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2answers
63 views

Lack of understanding of the proof of $\sum\limits_{i\in\mathbb{Z}}2^{-|i|}=3$.

In Elementary Real Analysis of Thomson and Bruckner p85 the proof of $\sum\limits_{i\in\mathbb{Z}}2^{-|i|}=3$ is given: I didn't understand why:$$\sum_{j\in J}2^{-|j|}<2(2^{-N})$$ Could you please ...
0
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2answers
36 views

Sandwhich/Squeeze theorem for (1/n)tan(1/n)

I need to use the sandwhich/theorem to show that: $\frac{1}{n}\tan\frac{1}{n}\to 0$ as $ n \to \infty $ I'm not sure how to do this. I understand how to use the theorem to prove that ...
0
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0answers
20 views

Is there any theorem on the convergence of area integral and L2 inner product

Suppose that $f$ is an unknown function on $R^d$ and $\hat f(x)=\sum_{i=1}^N w_i\delta(x-x_i)$ is a discrete approximation of $f$ given by some estimation method, where $\delta(x)$ denotes the Dirac ...
4
votes
2answers
73 views

prove that $\sum_{n=1}^{\infty}\frac{a_n}{n^\beta} $ converges.

If series $\sum\limits_{n=1}^{\infty}\frac{a_n}{n^\alpha} $converges, for any $\beta>\alpha$, prove that $\sum\limits_{n=1}^{\infty}\frac{a_n}{n^\beta} $ also converges. I suppose that it can be ...