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

learn more… | top users | synonyms (3)

2
votes
0answers
13 views

What is this sequence of polynomials?

i was trying to calculate the probability of something and i came upon them. i needed to know what this was equal to: $$p_n(x)=\sum_{k_n=k_{n-1}}^{x}....\sum_{k_3=k_2}^{x} \sum_{k_2=k_1}^{x} ...
0
votes
0answers
5 views

How to simultaneously search multiple finite sequences in OEIS?

Suppose I have several finite sequences and I want to know if they show up in a family of sequences or any one particular sequence or on rows of an array on OEIS; is there such a search option.
1
vote
5answers
58 views

$\sum\limits_{n=1}^\infty n(\frac{1}{2})^{n}$

I am trying to find the expected value of the number of even numbers rolled before the first odd number when rolling a fair die until an odd number comes up. I arrived at $\sum\limits_{n=1}^\infty ...
1
vote
2answers
26 views

Prove $\lim_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim_{n \to \infty} X_n}$, where $\{X_n\}_{n=1}^\infty$ converges

Let $\{X_n\}_{n=1}^\infty$ be a convergence sequence such that $X_n \geq 0$ and $k \in \mathbb{N}$. Then $$ \lim_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim_{n \to \infty} X_n}. $$ Can someone help ...
0
votes
1answer
27 views

Prove existence of $\{X_n\}_{n=1}^\infty$, $\{Y_n\}_{n=1}^\infty \subseteq S \subseteq \mathbb{R}$, $S \neq \emptyset$ and $S$ bounded

Let $S$ contained in $\mathbb{R}$ be a nonempty bounded set. Then there exists monotone sequences $\{X_n\}_{n=1}^\infty$ and $\{Y_n\}_{n=1}^\infty$ such that $X_n$ and $Y_n$ is contained in $S$. How ...
0
votes
2answers
45 views

Find the sum of the following series

Find the sum of the following series $$ \sum_{n=1}^\infty (-1) \frac{1}{n}\frac{9}{6^n}. $$ I think that $r$ is $\frac{9}{6^n}$ and $a$ is $-1$. But I'm not positive if I'm starting this problem ...
-1
votes
0answers
32 views

Calculated by multiplying the arithmetic progression terms

$$\prod_{k=1}^n(a+(k-1)d)=a\cdot(a+d)\cdot(a+2d)\cdot(a+3d)\cdots(a+(n-1)d)=\text{?}$$ Please help me! What is the formula?
3
votes
3answers
106 views

Help with radius of convergence of a power series.

I need to determine the radius of convergence of the series $\sum_{n=1}^\infty a_nx^n$, where $a_n=a^n+b^n$ and $a,b$ are real numbers. Not sure how to approach this one.
-1
votes
0answers
13 views

Finding sum of special series

How can you find the sum of this series. I got it in my test today, but I can't solve it (2^2/1*2)C0 + (2^3/2*3)C1 + .... + (2^12/11*12)C10 Where C0,C1... Are binomial coefficients
-1
votes
1answer
23 views

Explanation for sum of sequence

I saw that in a textbook. Could somebody explain how this sum of a sequence was obtained? ⌈n/2⌉+...+⌈n/2⌉+⌈n/2⌉ = ⌈(n+1)/2⌉⌈n/2⌉
1
vote
0answers
12 views

Combining even and odd parts of a Chebyshev series

I imagine this will be an easy problem, perhaps even routine, for some. I am learning to manipulate sums and need insight. I started with a power series $$s(x) = \sum_{n=0}^{\infty} a_n x^n$$ and ...
5
votes
1answer
37 views

Sequence Limit Problem: If $0 \leq x_{m+n} \leq x_n + x_m$ then limit of $x_n/n$ exists

If the sequence $\{x_n\}$ satisfies the property that $0 \leq x_{m+n} \leq x_n + x_m$ for all $n$, $m \in \mathbb{N}$ , show that the limit of the sequence $\left\{\frac{x_n}{n}\right\}_n$ exists. ...
4
votes
1answer
24 views

Divergent subsequence of an unbounded sequence

Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$. Working ...
0
votes
2answers
16 views

General Term of specific Alternating Series

Recently I think about series below and I wonder if there is away to write the general term of it... $$ ...
6
votes
0answers
70 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
1
vote
1answer
19 views

How to make a formula to link two sequences without any positioning numbers?

I have two sequences: sequence $n$: $4, 7, 10, 13$ sequence $p: 10, 16, 22, 28$ I wish to find a link between them i.e. $p=??$ However, I cannot use positioning numbers, so you can't say that $4$ ...
1
vote
4answers
64 views

Why do we assign values to divergent series? [duplicate]

Why do we assign values to divergent series? For example, the series $1+2+3+4... = -1/12$. I understand the proof for this, but I feel like it uses false math, and I recall reading that you can't do ...
0
votes
0answers
17 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
0
votes
0answers
11 views

Can the following product be written in the given form?

Can this: $$\eqalign{ & {\rm B}\left( {{m_3},{m_2} + m} \right)F\left( {m,{m_2};{m_3} + {m_2} + m;z} \right),where{\text{ }}{m_2},{m_3},z{\text{ are constant,}} \cr & {\text{m}} = ...
4
votes
1answer
90 views

a doubt with the series $ \sum_{n=0}^{\infty}e^{-nx} $

I know that the series is equal to $$ \sum_{n=0}^{\infty}e^{-nx}= \frac{1}{1-e^{-x}}$$ However, if I expand each exponential term into a Taylor series I get $$ \sum_{n=0}^{\infty}e^{-nx}= ...
1
vote
1answer
26 views

Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ [duplicate]

I am trying to compute the sum $\sum_{i=0}^{\infty} \frac{i}{2^i}$ which I know should be equal to $2$, but I cannot prove it. If I am not mistaken, it should be a arithmetico-geometric series ...
0
votes
0answers
23 views

relations between the root test and the ratio test

relations between the root test and the ratio test I know the theorem is correct if they are exist $$ \lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \lim\inf\limits_{n\rightarrow ...
0
votes
0answers
12 views

Prove that a certain sequence of polynomials is symmetric

Given $p_0(x,y,z)=1$, $p_{n+1}(x,y,z)=(xy+yz+zx)p_n(x,y,z+1)+z^2(p_n(x,y,z+1)-p_n(x,y,z))$. Prove that all $p_n(x,y,z)$ are symmetric polynomials.
8
votes
1answer
128 views

How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
3
votes
1answer
40 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
1
vote
1answer
59 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
0
votes
1answer
17 views

If $r_n\to r$ and $s_n\to s$, then $(r \star s)_M/M \to rs$.

I was going to ask this question, but I think I figured it out, so I thought I'd post my answer: In this question of mine, a user's answer makes the following claim: Suppose $r_n$ and $s_n$ are ...
2
votes
2answers
42 views

Very slow convergence of a particular series?

I've read that $$ \sum_{k=2}^{\infty} \frac{1}{k (\log k)^2} = 2.1097\ldots $$ However when I compute the partial sums it looks like a lot of terms are needed to even get the first decimals right. My ...
2
votes
2answers
68 views

Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
1
vote
1answer
87 views

Is $(1+2+3+…)=(1+2+2^2+2^3+…)(1+3+3^2+…)(1+5+5^2+…)…$?

Are these equal? $$(1+2+3+…)=(1+2+2^2+…)(1+3+3^2+…)(1+5+5^2+…)…$$ Where the RHS has a series for each prime. Looks like they are the same series by the fundamental theorem of arithmetic. Every number ...
3
votes
2answers
26 views

Another Divergent Series Question

Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from ...
0
votes
1answer
27 views

Divergent Infinite Series Question

If the infinite series $$\sum{a_n}$$ diverges where all terms are positive and $\lim{a_n}=0$, is it always possible to construct a subsequence such that its series converges? That is, does there ...
4
votes
0answers
58 views

Convergence of $\sum_n |\frac{\cos(3^n)}{n}|$

So a recent post asked about convergence of $\sum_n |\frac{\cos(2^n)}{n}|$, and using double-angle formula for $\cos$ it could be shown that for each pair of consecutive terms, at least one term had ...
2
votes
1answer
166 views

Does series converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do.
0
votes
2answers
40 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
0
votes
0answers
32 views

sequence and series of convergence problem

We have a sequence/series problem. and i need help Consider the following: How can we show that $2-2^{1/2}+2^{1/3}-2^{1/4}+2^{1/5}-2^{1/6}+\ldots$ diverges? I am so lost as to solving this problem. ...
4
votes
1answer
78 views
+50

Convergence of the series $\sum_{n=0}^\infty \frac{1}{n+1}\sin\bigr(\frac{p\pi u_n}{q}\bigl)$

Let $(u_n)_{n\in \mathbb{N}}$ defined by : $u_0=1, u_1=1$ and for all integer $u_{n+1}=3u_n-u_{n-1}$ Study the convergence of $$\displaystyle\sum_{n=0}^\infty ...
3
votes
1answer
56 views

Simple demonstration for $\lim_{n\to\infty}(\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}) = \frac{1}{e}$ [duplicate]

What is the simple demonstration with elementary means for Lalescu Sequence: $$\lim_{n\to\infty}(\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}) = \frac{1}{e}?$$ (Traian Lalescu-romanian mathematician (1882-1929))
1
vote
1answer
68 views

If $\sum\limits_na_n$ diverges and $(a_n)$ is positive decreasing then $\sum\limits_n\min(a_{n},\frac{1}{n})$ diverges [duplicate]

If the sequence $\{a_{n}\}$ monotonically decreases to $ 0$ and the series $\sum\limits_{n}a_{n}$ diverges, then the series $\sum\limits_{n}\min(a_{n},\frac{1}{n})$ diverges as well. my idea: ...
0
votes
0answers
22 views

how to generate a magic series?

Please explain the concept of magic series AND I want to generate a magic series for a given range of numbers. For instance, assume range = [0..10] for each index of the range I want a corresponding ...
2
votes
2answers
71 views

What is the sufficient and necessary condition for changing the order of summation?

What is the necessary and sufficient condition for $\sum\limits_{i=0}^{\infty }{\sum\limits_{j=0}^{\infty }{{{a}_{ij}}}}=\sum\limits_{j=0}^{\infty }{\sum\limits_{i=0}^{\infty }{{{a}_{ij}}}}$? Suppose ...
3
votes
1answer
31 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
2
votes
1answer
66 views

Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems its a 'simple' property, but I haven't been able to ...
0
votes
1answer
29 views

Product of two geometric series

I have used the Product of two power series and find out the below results. But it is to some extend strange for me, could you please confirm the results? Let $A=\sum_{i=0}^{\infty}(\frac{L}{a})^i$ ...
2
votes
4answers
83 views

How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $

I have put his on wolfram and obtained answer as follows: $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$ And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$ However I am ...
1
vote
1answer
8 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
1
vote
2answers
20 views

Few questions about limit of sequence

Question:Find the the limits of these sequence.if the limits does no exist ,explain why. (1)$\left \{ cos((2n+1)\frac{\pi}{2}) \right \}_{n=1}^{\infty }$ my answer: so when n=1,lim=0 (2)$\left \{ ...
1
vote
1answer
37 views

How to find the Fourier series of a periodic function

Find the Fourier series of the function $f(t)=3t^2$, $-1\le t\le 1$. How do I solve this problem? What is the general formula and the way to solve this?
0
votes
3answers
60 views

How does this series diverge?

The series: $$\sum_{n=0}^{\infty} \sqrt{n^2 +1} -n$$ diverges. Can someone please tell me how this is proven and done.
9
votes
3answers
538 views

Example of a sequence with more than one limit.

I have heard of the idea of a sequence converging to more than one limit, but I cannot imagine how it would work. Could someone give me an example of such a case, and explain how it works?