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

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1
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3answers
26 views

value of an $\sum_0^\infty\frac{3n-4}{(n-2)(n-1)n}$

I ran into this sum $$\sum_{n=0}^{\infty} \frac{3n-4}{n(n-1)(n-2)}$$ I tried to derive it from a standard sequence using integration and derivatives, but couldn't find a proper function to describe ...
-1
votes
4answers
98 views

Help finding the limit of this series $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots$

How can I go about finding the limit of $$\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots = \sum_{k = 1}^{\infty} \frac{1}{2^{n+1}}?$$ Could I use the absolute value theorem? I have a ...
3
votes
1answer
39 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
0
votes
1answer
35 views

Seemingly simple logic question

I found this pleasant textbook on Proof Theory online and free: Introduction to Proofs, an Inquiry-Based approach To quote (page 9): 2.26 DEFINITION. A sequence $<x_0,x_1, . . . ,x_{n-1}>$ ...
0
votes
2answers
30 views

Proving a Recursive Formula

I know there are some questions on this site about how to find a recursive formula, but I've already found the formula. I'm doing an assignment (http://mathstat.dal.ca/~svenjah/math2112/Assign6.pdf) ...
0
votes
2answers
42 views

Give the three numbers that form a geometric sequence.

Three numbers form a geometric sequence. If 5 is added to the second term, then the resulting numbers will constitute an arithmetic sequence. If 22.5 is added to the third number, these numbers will ...
1
vote
1answer
68 views

Sum of trigonometric infinite series

I am trying to prove that for any $x\geq 1$ we have: $$ \sum_{m=1}^{\infty} \frac{\sin\frac{(2m-1)\pi}{x}}{\left(\frac{(2m-1)\pi}{x}\right)^3} = \frac{x}{8}(x-1). $$ Could I have some help please? I ...
4
votes
4answers
44 views

finding $a_1$ in an arithmetic progression

Given an arithmetic progression such that: $$a_{n+1}=\frac{9n^2-21n+10}{a_n}$$ How can I find the value of $a_1$? I tried using $a_{n+1}=a_1+nd$ but I think it's a loop.. Thanks.
3
votes
2answers
190 views

How to prove the non-existence of a polynomial series that uniformly converges to this function?

I was asked to prove the following. There exists a function $p(x)\in C(a,b)$ $(-\infty<a<b<+\infty)$, such that there does not exists a polynomial series which uniformly converges to ...
0
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2answers
21 views

Excel's EXP function compared to a series expansion

I am comparing the results of a series expansion of $e^x$ to Excel's $\mathop{EXP}(x)$ function. Should I expect them to be the same? Excel's gives $\mathop{EXP}(10) = 22026.4657948067$. However, ...
0
votes
3answers
86 views

Cauchy convergent sequences

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample ...
60
votes
7answers
6k views

What's next in this number series? [closed]

340, 680, 1428, 3141.6, _____ This is from an aptitude test. I'm not able to find any pattern in them.
0
votes
1answer
35 views

By what rule can't you do this specific action with respect to infinite sums? [on hold]

An example of this is the summation of 1+2+3...=-1/12. By some reason, you cannot change the digits of that to 1+(1+1)+(1+1+1)... which would be equal to -1/2. -1/12 is not equal to -1/2 though.
3
votes
3answers
65 views

“fast enough” decay of an $\ell ^2$ sequence implies $\ell ^1$?

To be specific, say we are given that $(a_n)$ is a sequence of real numbers such that \begin{equation} \sum_{n=1}^{\infty} n^3 a_n^2 < \infty. \end{equation} Is it then true that $$ ...
3
votes
0answers
111 views

An interesting identity involving powers of $\pi$ and alternating zeta series

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
0
votes
3answers
85 views

Sum of $\sum\limits_{x=-\infty}^{\infty}x^{\operatorname{sign}(x)}$

Both the sum of $1+2+3+4+\cdots$ and the sum of $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots$ diverge. If both are paired together in one function, as seen above, can they amount to a ...
1
vote
1answer
38 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
1
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2answers
60 views

How to prove that $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$

I am currently trying to prove: $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$ I can easily squeeze the series between 0 and 1. I don't know many handy ...
4
votes
6answers
288 views

Evaluating $\lim_{n\to\infty} \frac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ using integral

Evaluate $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ This is the question I remembered from my high school textbook (I remembered it while reading about ...
4
votes
1answer
90 views
+100

Prove a sequence converges using sub-sequences

Let there be a sequence $a_n$ The following sub-sequences converge: $a_{n^3},a^3_{2n+3}-a^3_{2n+4},a^2_{2n+3}-a^2_{2n+4},a_{2n+15}$ Prove: $a_n$ converges I think it has something to ...
13
votes
2answers
395 views

Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\{A_n\}\in \mathbb{R}^{m\times m}$ be a sequence of symmetric matrices such that $A_n\to A$ as $n\to \infty$, i.e. $\lim_{n\to \infty}a_{ij}(n)=a_{ij}\ \forall 1\le i,j\le m$ where ...
3
votes
2answers
54 views

Find the sum$\pmod{1000}$

Find $$1\cdot 2 - 2\cdot 3 + 3\cdot 4 - \cdots + 2015 \cdot 2016 \pmod{1000}$$ I first tried factoring, $$2(1 - 3 + 6 - 10 + \cdots + 2015 \cdot 1008)$$ I know that $\pmod{1000}$ is the last ...
1
vote
2answers
20 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
3
votes
1answer
55 views

Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
60
votes
3answers
5k views

Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will ...
1
vote
1answer
100 views

Is $(\frac 1{n^2 \sin n })$ convergent ? If so , what is the limit? [duplicate]

Is the sequence $\left(\dfrac 1{n^2 \sin n }\right)$ convergent ? If it does, with what limit?
2
votes
1answer
78 views

How to calculate the $n$-th member of sequence $a_{n+1}=\sqrt{y+a_{n}}$

I was searching for a smooth continuous concave function $$f:R^{+}\times R^{+}\to R^{+}$$ so that $$f(x+1,y)=\sqrt{y+f(x,y)}\quad\text{and}\quad f(0,y)=0.$$ But I couldn't find a general function, ...
4
votes
0answers
57 views

Find the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$

I'm asked to prove that the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$ is the unit circle. My try: First, use root test to show that the series converges for $|z|<1$. Then I have ...
0
votes
2answers
26 views

suppose $a_n>1$ $a_n$is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}(1-\frac{a_n}{a_{n+1}})\frac{1}{\sqrt{a_{n+1}}}$ converges

suppose $a_n>1$, $\{a_n\}$ is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}\left(1-\frac{a_n}{a_{n+1}}\right)\frac{1}{\sqrt{a_{n+1}}}$ converges I don't have any idea about how to ...
0
votes
4answers
48 views

Proof of Convergence of a Sequence

Show that the sequence $\frac{n^2+1}{n^2+n}$ converges and its limit is $1$. However, I am finding it difficult to prove according to the rules that a converging sequence must obey, that is, sequence ...
0
votes
0answers
16 views

Convergence a series with non-negative terms and it's relationship with geometric series [on hold]

Suppose $\sum_{n=1}^{\infty} a_n$ is a convergent series of non-negative terms. 1, Does there then exist a $q \in (0,1)$ and $k \in \mathbb N$ , such that $a_n \leq q^n$ for all $n > k$ ? 2, Is ...
1
vote
0answers
18 views

Periodicity of Newton's method approximations on a cubic polynomial

Bruckner & Bruckner, Elementary Real Analysis Let $f(x) = x^3 - 3x + 3$ Applying Newton's method to get $x_{n+1} = x_n -\frac{f(x)}{f'(x)} \ ,$ prove that for any positive integer $p$, there ...
4
votes
1answer
32 views

Rationality of subseries

Let $(a_n)_n$ be a real sequence. Assume than $\sum_{n=1}^\infty a_n$ converges to a rational number. Does there exist a subseries which converges to an irrational constant? Assume now the opposite ...
-3
votes
1answer
35 views

For what values of α the following serie converges?

For what values of α the following series converges? $\displaystyle\sum_{n=1}^{\infty} (\frac{1}{n}-\sin\frac{1}{n})^{\alpha}$ Help.. Thanks...
0
votes
3answers
73 views

Proof the series is finite using following inequality

Let $$a_n=\frac1{\sqrt1}+\frac1{\sqrt 2}+\ldots +\frac1{\sqrt n}-2\sqrt n $$ For the task to prove that $$\tag1-2\le a_n\le -1 $$ I was given the hint $$\tag2\sqrt{k+1}-\sqrt k<\frac1{2\sqrt ...
3
votes
1answer
69 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}\left|\frac{nx^n}{S_{n-1}}-1\right|$$ I need to show that ...
0
votes
2answers
412 views

A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual…

A farmer buys a used tractor for Rs $12000$. He pays Rs $6000$ cash and agrees to pay the balance in annual installments of Rs $500$ plus $12 \%$ interest on the unpaid amount. How much will be the ...
1
vote
3answers
67 views

probability and expected value

Hey I am not sure if I thinking correctly on this question? In a carnival, there is game which charges you $3$ dollars to play a game. You win $1$ dollar for every consecutive head you get and you you ...
1
vote
1answer
33 views

Proving the closed form for an infinite sum (related to Chebyshev polynomials)

How do I prove the following identity? For $y\not= 0$, we have $$ \sum_{n=0}^{\infty} \dfrac{1}{2y}\left( (x+y)^{n+1}-(x-y)^{n+1}\right) = \dfrac{1}{(x+y-1)(x-y-1)}. $$ I am trying to find the ...
4
votes
2answers
68 views

$\sum_{n=1}^{\infty} \frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)$ diverges [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
-1
votes
2answers
74 views

Convergence of series $\sum^{\infty }_{n=2}\frac{\log[(1\text{+}\frac{1}{n} )^{n}(n+1)]}{\log(n)^{n}\log\text{(}n+1)^{n+1}} =\log_{2}\sqrt{e}$ [duplicate]

I need help with this problem, I'm very lost with the algebraic expression. I'd appreciate your opinions. $$ \sum^{\infty }_{n=2}\frac{\log[(1\text{+}\frac{1}{n} ...
0
votes
4answers
92 views

Limit of $\{a_n\}$, where $a_{n+1} = \sqrt{2+a_n}$ [on hold]

I am struggling with this question: Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. ...
3
votes
4answers
41 views

Difference of consecutive pairs of sequence terms tends to $0$

This seems an elementary problem, but I don't know of any reference to it in the literature. Consider the sequence $(a_n)_{n=1}^\infty$ of real numbers. Suppose ...
2
votes
2answers
43 views

Converge series such that permuting the termes will change the limit.

I know that for a series that converge, if we permute the element of the sum, the series doesn't necessarily converge. For exemple $$\sum_{n=1}^\infty \frac{(-1)^n}{n}$$ converge but if we first sum ...
7
votes
6answers
456 views

Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$

Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ ...
5
votes
2answers
353 views

Convergence of a sequence of real numbers

Let $\alpha, \gamma$ be real numbers such that $0<\alpha<1$ and $\gamma>0$. Consider the sequence of real numbers given by $$ \begin{cases} x_0\ne 0&\\ ...
0
votes
1answer
28 views

How do I formulate a specific formula for a sequence?

I have three arrays, for instance s = [1:2], j = [1:20] and b = [1:8], and I am trying to build a single row. The problem that I actually have is that I need to find a formula f(s,j,b) such that ...
6
votes
1answer
89 views

Convergence of $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$

I'm looking at some notes from my previous complex variables course and I need help verifying some things about the convergence of $$ \prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right) $$ on compact ...
0
votes
0answers
30 views

How to separate self-defining values from sigma?

$$\sum_{k=1}^{m} \sum_{j=1}^{n} a_kx_j^{b_k+b_i} = \sum_{j=1}^{n} y_jx_j^{b_i}$$ What I need to do is solve $a$ for every $i$ given ($i$ is between 1 and $m$), so their result won't be composed of ...