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

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0
votes
4answers
128 views

sequence $x_{n+1} = x_{n} + \sin x_{n}$

There is a sequence which satisfies $$x_{1} = a$$ $$x_{n+1} = x_{n} + \sin x_{n}$$ where a = 1 . Why does $$\lim_{n \rightarrow \infty} x_{n} = \pi$$ hold ?? (the first version of question was with ...
2
votes
1answer
51 views

Generalizing the Monotone Subsequence theorem

In proving the Bolzaono-Weierstrass theorem, one proves the lemma that every infinite real sequence has a(n infinite) monotone subsequence. In all of the proofs I've seen so far, this is done by ...
1
vote
3answers
41 views

Series Convergence/Divergence $\frac{n^n}{(n+1)^{n+1}}$

Trying to establish whether $\sum x_n$ for $x_n := \frac{n^n}{(n+1)^{n+1}}$ converges or diverges. Here's what I've done so far: 1) n-th term: $x_n < \frac{n^n}{n^{n+1}} = \frac{1}{n}$, so ...
1
vote
1answer
12 views

Find the first $3$ terms of the two possible geometric progressions.

The fourth term of a G.P is $3$ and the sixth term is $147$. Find the first $3$ terms of the two possible geometric progressions. Can you help me find $a$ and $r$? It is too complicated. I took two ...
0
votes
2answers
51 views

Brian Tracy 1000% Productivity Formula?

Hi I recently watched a Brian Tracy productivity video and I am curious about the maths behind this, I understand compound interest well but I cannot seem to get the numbers working for me. How does ...
5
votes
1answer
83 views

An inequality about a sequence

Let $(a_n)$ be a sequence such that $a_0=1 , a_1=2 , a_{n+1}=a_n+\dfrac {a_{n-1}}{1+ a_{n-1}^2} , \forall n \ge1 $ , then is it true that $52 < a_{1371} < 65$ ? $ EDIT:-$ I am posing another ...
0
votes
2answers
22 views

Series Ratio Test Convergence

$\displaystyle\sum_{k=1}^\infty (2k)!/k!(k+1)!$ Let $a_k = (2k)!/k!(k+1)!$ $\lvert a_{k+1}/a_k\rvert \to 4$ as $k \to \infty$ Thus the series is divergent. Can someone double check ... my gut says ...
0
votes
1answer
41 views

Find the term of the G.P.

Find the term of the G.P. $1, 1.2, 1.44, 1.728,... $ which is just greater than $100$. Please somebody explain me the question? All i know is $a=1$ and $r=1.2$ Help :( should it be $T_{n} > ...
1
vote
1answer
17 views

A question about Abel's test

Abel's test for $\sum a_n b_n$ to converge requires: $\sum a_n$ converges ${b_n}$ is bounded ${b_n}$ is monotone. My question is why do we need the 3d condition? The 2nd condition ...
-2
votes
1answer
97 views

infinite sum of alternating series $\sum_{n = 1}^{\infty} (-1)^{n + 1} \frac{n^2}{1 + n^3}$ [closed]

please help me for the solution of \begin{align*} \sum_{n = 1}^{\infty} (-1)^{n + 1} \frac{n^2}{1 + n^3} &= \frac 1 3 \left[1 - \ln 2 + \pi \operatorname{sech}\left(\frac{\sqrt 3}{2} ...
1
vote
1answer
21 views

Find the sum of the 10th and 11th terms of the G.P.

The third term of a geometric progression of positive terms is $\frac{6}{25}$ and the seventh term is $1\frac{23}{27}$. Find the sum of the 10th and 11th terms of the G.P., giving your answers correct ...
0
votes
2answers
59 views

Proving Alternating Series Convergence

Suppose $x_n > 0$ and $\sum_{n=0}^\infty x_n$ is convergent. Prove that $\sum_{n=0}^\infty (-1)^nx_n$ is convergent. Any hints or starting points? So far I figured that I should show that the ...
0
votes
0answers
12 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
1
vote
0answers
38 views

Alternating Series Test for convergence [closed]

Trying to figure out if the following alternating series converges or diverges. Using the alternating series test (AST) criteria, the series appears to converge as the lim of 1/nlnn is 0 and the ...
5
votes
3answers
71 views

Convergent Sequence, mean of previous three numbers

I am given three real numbers $x_0, x_1, x_2$ and the next number in the sequence is defined as the mean of the three real numbers. So: $$x_a=\tfrac{1}{3} \cdot (x_{a-3} + x_{a-2} + x_{a-1})$$ I am ...
1
vote
0answers
47 views

Alternating Series Divergence

Test the series for convergence: $$ -2/5 + 4/6 - 6/7 + 8/8 - 10/9$$ Attempted Solution: $$a_n = (-1)^n,\; b_n = \frac{2n}{4+n}$$ $$b_n\not\stackrel{n\to\infty}{\longrightarrow}0\implies \sum_n ...
3
votes
1answer
108 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
4
votes
5answers
226 views

How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$ \sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)} $$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
3
votes
1answer
44 views

$\sum_{n=0}^{\infty} a_n x^n$ and $\sum_{n=0}^{\infty} a_{n^2} x^n$ with different radii of convergence

Could you give an example of $$\sum_{n=0}^{\infty} a_n x^n$$ and $$\sum_{n=0}^{\infty} a_{n^2} x^n$$ that have different radii of convergence?
1
vote
1answer
36 views

Convergent strictly increasing sequence $a_n$ $\Rightarrow$ sequence $f(a_n)$ is convergent.

Strictly increasing $f(x)$ is defined on R. Then for any convergent strictly increasing sequence $a_n$ sequence $f(a_n)$ is also convergent. The answer is TRUE, but I believe I have a ...
1
vote
0answers
28 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
1
vote
0answers
62 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
4
votes
2answers
122 views

Limit of $a(k)$ in $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $

For n = 1, 2, 3 ... (natural number) $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $ $ a_1 = 1, \ a_2 = \frac{1}{2}, \ a_3 = \frac{7}{12} \cdots $ What is the limit of {$ a_k $} $ \lim_{k \to \infty} ...
8
votes
0answers
310 views
+100

Does this sequence have any mathematical significance?

Take the sequence 001 and repeatedly append its second half to itself, using the larger half if the length is odd. This gives you ...
2
votes
2answers
97 views

Complex series radius convergence

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ...
0
votes
0answers
24 views

Two case about convergent series

Could you help me to prove analytically that ? I started to study Taylor Series and I'm lost. Thanks in advance.
0
votes
1answer
35 views

Solving a ODE using series

I have to prove that the series $$y(x)=\sum_{n=0}^{+\infty}\frac{x^{n}}{(n!)^{2}}$$ satisfies the ODE $$xy''+y'-y=0$$ When I derivate and substitute in the equation, I get ...
9
votes
8answers
252 views

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

Question: Proof that if $$a_n=\left\{ \begin{array}{ll} a_1=1\\ a_{n+1}=1+\frac{1}{1+a_n} \end{array} \right.$$ then $a_n$ converge, and then find $\lim_{n \to \infty}a_n$. I found ...
2
votes
3answers
101 views

Calc 2: convergent of divergent sequences

I would like to know if this sequence, $\sin\left(\frac{n \pi}{2}\right)$ ,is convergent or divergent? I have done this problem and I know that it is divergent through oscillation (I'm pretty sure). ...
1
vote
2answers
33 views

Test for convergence with either comparison test or limit comparison test

Tried using $b_n = \frac1{n^n + 1}$ with limit test which indicated that both either converge or diverge but getting stuck on how to show that one actually does converge.
0
votes
1answer
35 views

Is there an explicit formula for this homomorphism?

Set $A$ is a finite set of whole numbers from $1$ to $n$ (for some arbitrary $n$). Constant $c$ is an arbitrary whole number. I want to partition the set into $c$ ordered groups, so that each ordered ...
1
vote
2answers
84 views

Test for convergence $\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$

Test for convergence $$\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$$ So far I attempted to use the ratio test, but I'm stuck on what to do after. ...
0
votes
3answers
41 views

Use the limit comparison test on the following series: $\sum_{n=1}^\infty\frac{5n^3+1}{2^n(n^3+n+1)}.$ [closed]

Use the limit comparison test on the following series. Can't figure out a good bn to simplify this problem. $$ \sum_{n=1}^\infty\dfrac{5n^3+1}{2^n(n^3+n+1)}.$$
1
vote
1answer
56 views

Characterizing the primes which don't divide any Pell-Lucas number(s)

For integer $n$, let $P_n$ be a Pell number, and $Q_n$ its companion. Is there a characterization of the prime numbers $p$ which don't divide any $Q_n$? By brute-force search, I found that this ...
0
votes
0answers
19 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
2
votes
2answers
43 views

if $a>1$, Prove that $\lim a^{1\over n}=1$

if $a>1$, Prove that $\lim a^{1\over n}=1$ Is the result true if $0<a\le1 ?$ My attempt : let $a^{1/n}=1+h$, then $a=1+nh+\frac{n(n-1)h^2}{2}+\dots+h^n$ so, $a>\frac{n(n-1)h^2}{2}$ or, ...
0
votes
2answers
62 views

What number does not belong to the following series? [closed]

I was doing an IQ test with my friends, when I found a question that left me stumped: What number does not belong in this series? 2 - 3 - 6 - 7 - 8 - 14 - 15 - 30 It is only one number, and I ...
12
votes
1answer
144 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
1
vote
1answer
33 views

Comparison test involiving e

Just one thing: with each sum can I compare $$a_{n}=e-\left(1+\frac{1}{n}\right)^{n}$$ to prove that the series $$\sum a_{n}$$ diverges?
1
vote
0answers
54 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
2
votes
2answers
50 views

Series convergence - Gauss test

How do I prove that $$\sum_{n=1}^\infty\left(\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot(2n)}\right)^{k}$$ converges for $k>2$ using Gauss test?
0
votes
2answers
36 views

Sum of this series.

I tried manipulating it to get it into a binomial expansion of two known terms, but i seemingly failed. Please help me out. $$S=\displaystyle\sum_{r=0}^{12} \binom{12}{r} \cos \frac {r\pi}{6}$$
12
votes
3answers
205 views

How to show that $ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $

How to show that $$ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $$ where $e = \lim \left({1 + \dfrac 1 n}\right)^n$ I'm guessing this can be done using the Squeeze Theorem by applying the AM-GM ...
0
votes
1answer
14 views

Sequence of Partial Sums for repeated decimal

I have been trying to figure out an explicit formula for the sequence of partial sums of a repeating decimal. Take 0.09 repeating for example. Using the fact that it is a geometric series with r < ...
2
votes
1answer
44 views

Series for reciprocal of polylogarithm?

The series representation of a polylogarithm of order $s$ is given by $$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$ Are there any simplified expressions for $\dfrac{1}{\text{Li}_s(z)}$? ...
1
vote
1answer
26 views

Two cases involving Maclaurin Series

Could you help me to prove it? I'm working hard in it, but I got nothing.
12
votes
3answers
232 views

How to prove that $ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $?

Mathematica tells me that $$ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $$ I could prove it for $a\rightarrow 0$, $a=1$ and $a\rightarrow \infty$, but could not find a general ...
1
vote
2answers
179 views

Solution to curious infinite series

How exactly does one find a closed form to: $$ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]$$ When expanded it takes on the form $$1 + ...
2
votes
3answers
34 views

Even-Odd pair in a sequence

Suppose we have a sequence of $n$ integers not necessarily distinct. Let's define, $E$ = Number of pairs $(i, j)$ such that $i<j$ and $A_i+A_j$ is even. $O$ = Number of pairs $(i, j)$ such that ...
1
vote
3answers
31 views

Geometric Sequences

Find geometric progression if $a_1 = 3$, $S_n = 2343$, $a_n = 1875$. I'm trying to use sum formula $S_n = a_1\dfrac{1-r^n}{1-r}$, but can't do much. I'm a bit lost so if anyone could help me.