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

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2
votes
2answers
37 views

If the sum of $n$ terms of an A.P. is $2n+3n^2$, find the $r^{th}$ term.

If the sum of $n$ terms of an Arithmetic Progression is $2n+3n^2$, find the $r^{th}$ term. Note: This question is from the book Higher Algebra by H.S. Hall & S.R. Knight and its answer is ...
-1
votes
4answers
130 views

Arithmetic Progression-Question from Hall and Knight's Higher Algebra

The question says- Between two numbers whose sum is $\frac{13}{6}$, an even number of arithmetic means is inserted such that the sum of these means exceeds their number by unity. How many means ...
0
votes
1answer
34 views

Accumulation point is a limit of some sequences in first-countable space with Axiom of Choice.

I have a statement which says that Let $X$ be a first-countable space and $E$ be a subset of $X$. If $p$ is an accumulation point of $E$, then there exists a sequence in $E$ that converges to ...
4
votes
1answer
78 views

Help in finding the sum of the series

$$\sum_{n=1}^\infty \frac{1}{n^4+n^2+1}$$ I tried breaking into factors but it is not telescoping. $$\frac {1}{(n^2+n+1)(n^2-n+1)} = \frac {1}{2n} \left(\frac {1}{n^2-n+1} - \frac ...
2
votes
1answer
69 views

double root and newton method, a problem on solved exercise? [on hold]

$f(x)$ in $x= \alpha$ has double roots and define in $\alpha$ neighbor. if the sequence $\{x_n\}$ for solving $f(x)=0$ calculated by newton methods the following is correct. ($a$ and $b$ is plasced ...
0
votes
1answer
46 views

Alternating series error estimatation theorem for $\sum_1^{\infty} (-1)^{n+3}\frac{n}{n^2+8}$

$$\sum_1^{\infty} (-1)^{n+3}\frac{n}{n^2+8}$$ Here is my question: Using the alternating series error estimation theorem, what is the smallest number of terms needed to estimate the entire sum with ...
5
votes
2answers
209 views

recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I've been trying to solve this recurrence relation for a week, but I haven't come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first ...
2
votes
1answer
36 views

Divergent series of independent RV

I'm trying to prove that if $\{X_n\}_{n=1}^{\infty}$ is a sequence of independent random variables with the same distribution and $P(X_1 \neq 0)>0$, then the series $\sum_{n=1}^{\infty} X_n$ is ...
-2
votes
2answers
34 views

Explain why the following sums of a harmonic series is greater than or equal to 1/2. [closed]

The (non-geometric) series $\frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \cdots$ is called the harmonic series. a) Explain why each of the following sums is greater than or equal to 1/2. ...
-2
votes
1answer
62 views
1
vote
1answer
70 views

evaluate trigonometric

To all the genius out there , here is a question about expresssing summation of hyperboilc functions : First of all, I've already proved that: $$\sinh(x + 1)- \sinh(x) = (-􀀀1 + \cosh (1)) \sinh(x) ...
2
votes
2answers
78 views

Prove or Disprove: If $a_n$ was positive and its limit is positive then the limit of the $\sqrt[n]{a_n}$ is $1$

If $a_n$ was positive and its limit is positive then the limit of the $\sqrt[n]{a_n}$ is $1$. I think it has to do with multiplication to start here,but I couldnt figure it out.
0
votes
1answer
34 views

Why do (the ranges of) these sequences intersect?

Let $\{(a_n,b_n)\}$, ($1\le n\le N$) be a finite sequence and $\{(s_n,t_n)\}$ ($n\ge 1$) be an infinite sequence, both in $(\{0\}\cup \mathbb{Z}^{+})^2$. We have $a_1=0$ and $b_N=0$. Also, either ...
1
vote
1answer
39 views

Prove or disprove: if the limit of $a_{n}$ goes to $\infty$, then $a_{n}$ has a minimum number.

Prove or disprove: if the limit of $a_{n}$ goes to $\infty$, then $a_{n}$ has a minimum number. Well I know from the definition of a series that goes to $\infty$ that there is a number $K>0$ that ...
1
vote
2answers
33 views

Problem with limit $\lim_{n\rightarrow\infty} |\lambda_{i}|^{n} n^{\ell}$

How I can calculate next: $$ \lim_{n\rightarrow\infty} |\lambda_{i}|^{n} n^{\ell} ? $$ Where $|\lambda_{i}|<1$ and $\ell$ is any positive integer.
0
votes
2answers
32 views

Is $\lim_{n\rightarrow\infty }nz^{n!n}=0$ for $|z|<1$?

Is $\lim_{n\rightarrow\infty }nz^{n!n}=0$ for $|z|<1$? We have a $\infty \cdot 0$ case, then how we proceed? How to use the L'Hospital's Rule? Thanks in advance!
0
votes
1answer
56 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...
0
votes
0answers
10 views

Approximating an Appell series with a second-order polynomial

Taking this Appell F1 hypergeometric series $$ f(t)=F_1\left(2;\frac{3}{2} (1-m),\frac{3}{2} (\lceil r\rceil +m-1);3;\frac{\frac{r}{2} t^2 \left(1-\frac{r}{\lfloor r\rfloor +m}\right)+\lfloor r\rfloor ...
2
votes
4answers
39 views

How can I find if $\sum_{n=2}^\infty {(-1)^n*4 \over (ln(n))^2} $ converges or diverges using the alternating series test?

$$4\sum_{n=2}^\infty {(-1)^n \over (\ln(n))^2} $$ If $4 \over (\ln(n))^2$ = $u_n$, then $u_n$> 0, and: $$\lim_{n\to\infty} u_n = 0 $$ But there is one more test to prove convergence which says ...
-1
votes
4answers
59 views

Root test for $\sum_{n=8}^\infty \left(1+\frac{3}{n}\right)^{n^2}$

I got that it would be infinity and diverge, but my answer seems to be incorrect. What did I do wrong? $$\lim_{n\to\infty}\left(1+\frac3n\right)^{n^2}= \lim_{n\to\infty} ...
2
votes
1answer
75 views

Showing that $\sum\limits_{n=1}^{\infty} (a_1+2a_2+…+na_n)/n(n+1) = \sum\limits_{i=1}^n a_n $

Let $\sum\limits_{n=1}^{\infty} a_n$ a series of positive terms convergent. Show that $\sum\limits_{n=1}^{\infty} \frac{a_1+2a_2+...+na_n}{n(n+1)}$ converges to the same value of $\sum\limits_{i=1}^n ...
1
vote
4answers
88 views

Explicit formula for recursive geometric/arithmatic series

In my Algebra 2 class, we have come upon a question that the class could not solve, and that the teacher has neglected to remove from the given packet for several years because of this. The problem is ...
0
votes
0answers
21 views

limit of recurrence serie by derive

how to define limit this series $$U_{n+1} = {df(U_n)\over dU_n}$$ $f(x)$ function derivable in Class $C^n$, $f^{(n)}(x)$ function not null and $U_0 \gt 0$
0
votes
0answers
24 views

True/false statements about convergence of a sequence

Let $(a_n)$ be a sequence. True or false: If $\lim_{n\to\infty}(a_{2n}-a_n)=0$ then $(a_n)$ converges. If $(a_n)$ converges then $\lim_{n\to\infty}(a_{2n}-a_n)=0$ Intuitively, I ...
-1
votes
2answers
126 views

A limit which is related to a series?

How to compute the following limit which is related to a series? $$ \lim_{N\rightarrow\infty}N^2\sum^{N-1}_{k=1}\left(\frac{k}{N}\right)^{N\ln N}$$
1
vote
2answers
178 views

Uniform convergence of the series for $\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$

I am looking for the values where this series expansion converges uniformly. $$\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$$ Intuitively, I believe ...
0
votes
1answer
56 views

How simplify this sum?

I need help to simplify this sum : $$\sum_{i=0}^{x-1}\left(1-\dfrac{1}{2^i}\right)^{m-1}$$ Is it possible to simplify it ? Thank you.
22
votes
5answers
420 views

How can you derive $\sin(x) = \sin(x+2\pi)$ from the Taylor series for $\sin(x)$?

\begin{eqnarray*} \sin(x) & = & x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\\ \sin(x+2π) & = & x + 2\pi - \frac{(x+2π)^3}{3!} + \frac{(x+2π)^5}{5!} - \ldots \\ \end{eqnarray*} Those ...
3
votes
3answers
95 views

Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?

I am interested to find $$\sum_{k=1}^n\frac{1}{k(k+1)}$$ without using telescoping series method. I tried very hard but still could not think of a way to find it without using telescoping series ...
2
votes
1answer
61 views

How to evaluate $\sum\limits_{n=1}^\infty (-1)^{n-1} \ln (1+\frac1n)$

Can someone help me evaluate the sum of this series through elementary means? $$\sum_{n\geq 1}(-1)^{n-1} \ln \left(1+\frac1n\right)$$
2
votes
1answer
89 views

Almost sure convergence of the series of independent random variables

Let $\{X_n:n\ge1\}$ be i.i.d. random variables with $\operatorname EX_1=0$ and $\operatorname E|X_1|^p<\infty$, where $1<p<2$. Let $\{b_n:n\ge1\}$ be a real sequence. Does the series $$ ...
1
vote
2answers
802 views

How to find the pattern?

I'm challenging myself to figure out the mathematical expression of the number of possible combinations for certain parameters, and frankly I have no idea how. The rules are these: Take numbers ...
4
votes
2answers
141 views

Nature of the serie $\sum\prod_{k=2}^n (2-e^{\frac{1}{k}})$

I'd like to determine the nature of the following serie : $$\sum_{n\ge 2}\prod_{k=2}^n (2-e^{\frac{1}{k}})$$ Let $u_n = \prod_{k=2}^n (2-e^{\frac{1}{k}})$. So I "have": $$\begin{aligned} \ln(u_n) ...
0
votes
0answers
34 views

Calculating Infinite Sums [closed]

How do you calculate sum of infinitely sequenced numbers in an arithmetic or geometric progression? Please help me out! Thanks in advance!
3
votes
2answers
97 views

Prove $ \sum \frac{\cos n} { \sqrt n}$ converges

How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ? I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number } How to proceed ?
3
votes
2answers
47 views

Multiple part problem concerning the proof that $\sum_{k=1}^n k^3=\left(\frac{n(n+1)}{2}\right)^2$ by induction

So I'm having trouble with $c,d$ and $e$. For $c$ so far I have: Inductive Hypothesis: $(\frac{n(n+1)}{2})^2 = (\frac{(k+1)(k+2)}{2})^2$ is that correct?
4
votes
4answers
75 views

The sequence $\frac{2}{2-u_n}$ diverges

Let $(u_n)$ be a sequence defined with $u_{0}$ a real number such that $u_0 \notin \{0,1,2\}$ and $$u_{n+1} = \frac{2}{2-u_n}$$ Prove that $(u_n)$ diverges. I try to use the fact that this sequence ...
0
votes
2answers
34 views

Could someone help me clarify the steps for this solution?

Given $$\sum_{n=1}^\infty \frac{1}{n^6} = \frac{\pi^6}{945},$$ calcuate $$\sum_{n=1}^\infty\frac{1}{(n+2)^6}.$$ Solution: $$\sum_{n=1}^\infty\frac{1}{(n+2)^6} = \frac{1}{3^6} + \frac{1}{4^6} + ...
0
votes
0answers
43 views

Series verification

Anyone can tell me what series is this? As I heard that this kind of series already been long understood. I am required to calculate the value of $P_2$ from the 1st sequence, the value of $P_2$ is ...
2
votes
1answer
35 views

convergent series and divergent series

Hi I have two questions. First, $\sum_{n=1}^\infty \frac{n}{n^3+1}$. Is it divergent or convergent? I think it seems like it is positive and decreasing function so we can apply integral test. ...
1
vote
1answer
37 views

Derive an exact formula (solve the recurrence definition) for the following recursive sequence:

Derive an exact formula (solve the recurrence definition) for the following recursive sequence: $s_n = 2_{s_n-1} - s_{n-2}$ where $n \ge 2$, and $s_0 = 4$, $s_1 = 1$. So I saw someone solving this by ...
3
votes
2answers
64 views

How can I find the convergence/divergence of $\sum_{n=1}^\infty {n! \over n^n}$

Using either root test or ratio test. I have the feeling that it is the root test, I'm not sure how to proceed from this: $$ \sqrt[n]{n! \over n^n}= {(n!)^{1\over n} \over n} $$
1
vote
4answers
39 views

Proving that $\lim_{n\rightarrow\infty}n(a_{n+1}-a_{n})=1 \implies a_n$ diverges to $\infty$

I'm trying to prove that given a sequence $a_{n}$ such as $\displaystyle\lim_{n\rightarrow\infty}n(a_{n+1}-a_{n})=1,$ then $a_{n}$ diverges to $\infty.$ I'm lost searching a path to prove it. I ...
1
vote
3answers
71 views

How can I find if $\sum_{n=1}^\infty {n! \over 10^n} $ converges or diverges?

$$\sum_{n=1}^\infty {n! \over 10^n} $$ I wasn't sure on which method to use, I think the ratio test might work, but I'm stuck. Here's what I have so far: $a_n$= $n! \over 10^n$ & ...
4
votes
1answer
108 views

Sequence with Prime Numbers

I was looking a question in a calculus book which used the following steps to show that following sequence has a limit (called Euler's constant $\gamma$): $$t_n = \sum_{i=1}^n\left(\frac{1}{n}\right) ...
1
vote
1answer
55 views

Infinite series $\sum_{n=1}^{\infty}nx^{n+1}$ does not comply to any of my (known) tests

I am attempting to find the interval of convergence for $$\sum_{n=1}^{\infty}nx^{n+1}$$ The lower bound, x = -1, would be tested by determining if $$\sum_{n=1}^{\infty}n(-1)^{n+1}$$ diverges. ...
1
vote
1answer
40 views

Proof that $\lim \frac{a_n}{1+a_n^2} = 0 \implies \lim a_n = 0$

I´ve tried some exercises about sequences convergence, particularly: Let $a_{n}$ be a sequence such as $\displaystyle\lim_{n\rightarrow\infty}\frac{a_{n}}{1+a_{n}^2}=0.$ Prove that $a_{n}$ ...
0
votes
1answer
503 views

Prove that absolute convergence implies unconditional convergence

In the proof of "absolute convergence implies unconditional convergence" for a convergent series $\sum_{n=1}^{\infty}a_n$, we take a partial sum of first $n$ terms of both the original series ($S_n$) ...
1
vote
2answers
36 views

Prove that every natural number is a limit of a subsequence

I need to prove that every natural number is a partial limit (limit of a subsequence) to the sequence ${{a}_{n}}=n-{{\left\lfloor \sqrt{n} \right\rfloor }^{2}}$ . I already found that 0 is a partial ...
2
votes
2answers
53 views

How to find convergence/divergence of this series

$$\sum_{n=1}^\infty {1+\cos(n) \over n^2}$$ I used the comparison test and said that $\sum_{n-1}^\infty {1 \over n^2}$ is comparable and also larger than $\sum_{n=1}^\infty {1+\cos(n) \over n^2}$, ...