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

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8 views

Calculus scenario involving instantaneous and speed (sequences)

The scenario is nearly always the same as Wilie is standing at the end of a road that is 1 kilometer long, and there at the other end is that Roadrunner, he’s just standing there, sticking his tongue ...
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5answers
19 views

How do I find the radius and interval of convergence of $\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $

$$\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $$ I used the ratio test to test for absolute convergence, but I'm sort of stuck on: $$n(x+2) \over n+1$$
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2answers
47 views

A Sequence That has No Upper Bound But Does Not Tend To Infinity

Let $a_n$ be a sequence which has no upper bound. Give an counterexample sequence for the statement $$\lim_{n\to\infty} a_n=\infty$$ Any hint?
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1answer
24 views

Evaluating the sum of a partial geometric sequence using Sigma notation

I have a worksheet from my instructor with this problem on it, but the solution he has given is different from what I got, and I don't know why. I'm not sure how to input the Greek letter sigma, but ...
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0answers
15 views

Naive Bayes' classifier

Here's the problem set: I got the first two sections down but I have no idea how to do the third section. Can anyone help?
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5answers
26 views

Proving convergence/divergence via the ratio test

Consider the series $$\sum\limits_{k=1}^\infty \frac{-3^k\cdot k!}{k^k}$$ Using the ratio test, the expression $\frac{|a_{k+1}|}{|a_k|}$ is calculated as: $$\frac{3^{k+1}\cdot ...
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1answer
17 views

Proof of Convergence Based on Monotonicity and a Limit

Theorem: Let the series $a_n$ be monotonous upward and $b_n$ be bounded. Let $\lim\limits_{n\to \infty}{a_n-b_n}=0$, then $b_n$ converges. I managed to prove by negation that $a_n$ is bounded and ...
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2answers
137 views

Find the sum of a series

I'm trying to find the sum of the following series: $$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n}$$ I tried to "convert" it to a simple geometrical series, but with no luck. Has someone any idea? ...
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2answers
35 views

How find $a \in (0, \pi)$ such that $(\cos(2^n a))_{n \ge 1}$ is convergent? [on hold]

Let $a \in (0, \pi)$ such that $(\cos(2^n a))_{n \ge 1} $ is convergent. How find $a$?
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0answers
14 views

How to show the series of expectations for truncated symmetric random variables is convergent

Suppose that $(X_n)$ is i.i.d. with symmetric distribution and that $E(|X_1|)<\infty$. I want to show that $\sum\limits_{i=1}^{\infty} \frac1iE(X_i 1_{[|X_i|<i]}) $ converges. Attempt: Since ...
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1answer
26 views

Formula for the sum of a geometric series

I'm using a book for my AS Level maths which says that: "The general rule for the sum of a geometric series is $$S_n = a\frac{r^n-1}{r-1}$$ or $$S_n= a\frac{1-r^n}{1-r}$$ " Why are there two ...
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2answers
28 views

How would I find power series $\sum_{n=0}^\infty {(3^nx^n) \over n!} $ radius and interval of convergence

$$\sum_{n=0}^\infty {(3^nx^n) \over n!} $$ I have no idea how to start this problem, the only thing that looks familiar is $x^n/n!$ which I know as a sequence goes to 0 when you take the limit, but I ...
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1answer
26 views

How prove that $0 < a_{10} - \sqrt{2}< 10^{-370}$ for $a_n = \frac{1}{2}(a_{n-1} + \frac{2}{a_{n-1}})$?

Let $a_1=1,$ , $a_n = \frac{1}{2}(a_{n-1} + \frac{2}{a_{n-1}})$. How prove that $0 < a_{10} - \sqrt{2}< 10^{-370}$? $a_n - a_{n-1} = \frac{1}{2a_{n-1}} \left(2 - a_{n-1}^2\right) < 0 ...
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4answers
31 views

How can I find the radius and interval of convergence of $\sum_{n=1}^\infty {(3x-2)^n \over n} $, and for what value x would it converge to?

$$\sum_{n=1}^\infty {(3x-2)^n \over n} $$ Not sure where to start with this problem. I'm thinking the ratio test because the numerator is raised to n, but n is also in the denominator.
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2answers
33 views

Taylor series of $\ln x$ at $x=e$

Like in the title, I need to find taylor series of $\ln (x)$ at $x=e$ I was thinking about changing $\ln (x)$ to $\ln (x-e+e)$ but it lead me to nowhere.
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0answers
9 views

Degree of a Sequence of Squares Recurrence Relation

We are told that there exists a k-th order homogeneous linear recurrence relation $a_n=r_1a_{n-1}+...+r_ka_{n-k}$ which has distinct roots. We need to prove that for every $a_n$ that is satisfied by ...
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0answers
27 views

Hilbert's double series theorem $\sum_{n,m}\frac{a_n b_m}{m+n}$ and its generalizations.

Let $l_p$ be the space of all complex sequences $x=\{x_n\}$ equipped with the finite norm $\|x\|_p=\left(\sum_n |x_n|^p\right)^{1/p}$. Hilbert's double series theorem states that $$\sum_{n,m}\frac{a_n ...
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1answer
16 views

Limit of sequence of analytic functions

If $\Omega_1$ and $\Omega_2$ are two nonempty disjoint open subsets ${\bf C}$ and $\{f_n\}$ is a sequence of analytic functions from $\Omega_1 \to \Omega_2$ which converges pointwise to a function $f ...
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4answers
122 views

Does $\sum 3^{-\sqrt{n}}$ converge or diverge?

I need to find out whether this series converges or diverges: $$\sum_{n=1}^\infty \frac 1{3^{\sqrt{n}}}$$ The $n$th term, ratio, and root tests are inconclusive, Abel's test doesn't apply (or I ...
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1answer
27 views

Manipulating series to find the recursive formula

Ok so I am stuck. I need to get all the $n$'s to $=0$ but I can't reduce my series which has $n=2$ to $0$ because then I will have undone all my work in the first place to get all the $X^n$'s to the ...
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1answer
32 views

Help please with finding the equation and pattern of Taylor Series. (2 problems I have attempted down below).

I didn't want to ask twice so I combined both of my questions together. I have just started on Taylor Series, and I'm not very good at figuring out patterns. First Question Find Taylor Series for ...
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2answers
29 views

How can I find the radius and interval of convergece of $\sum_{n=0}^\infty {(x+5)^n} $, and for what value of x does the series converge?

$$\sum_{n=0}^\infty {(x+5)^n} $$ We talked about this briefly but I'm still pretty confused about how to start this problem.
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2answers
51 views

Find the smallest $N$ such that $\sum_{k=1}^N\frac{1}{p_k}>\pi$. (The $p_k$'s are the prime numbers.)

How to solve the following problem? Let $\{p_k\}_{k=1}^\infty$ be the set of primes (in increasing order). What is the smallest integer $N$ such that $$\sum_{k=1}^N \frac{1}{p_k}>\pi?$$ We ...
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1answer
32 views

Test $\sum\limits_{n=1}^{\infty} \frac{(-1)^n\ln(n)}{n}$ for convergence

$\sum\limits_{n=1}^{\infty} \frac{(-1)^n\ln(n)}{n}$ The first thing I think to do is the alternating series test, but $\frac{\ln(n)}{n}$ is not a monotonically decreasing sequence. For example, the ...
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0answers
28 views

What to do when Ramanujan summation diverges too?

While using Ramanujan summation to some kind of divergent series I got stuck: let's take the definition of this sum for the terms of a general function $f(x)$: $$\Re(x)=\int_n^xf(t)dt-\frac ...
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0answers
27 views

An inequality of the type $(\text{const.})\sum_{i=1}^\infty b_i^2\leq\sum_{i=1}^\infty \left( \sum_{j=1}^\infty a_{i,j} \right)^2$

Let us consider a sequence of real numbers. It is known that $$\sum_{i=1}^N a_i^2 \leq 4\sum_{i=1}^N \left( \sum_{j=1}^i a_j \right)^2\ \ \ (*)$$ I have a curiosity. If we have a double series: ...
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1answer
28 views

Proving a sequence is convergent and calculating its limit

In my assignment I have to solve the following question. I think I have an idea how to solve it, but I suspect there is a little thing in my solution which is wrong. If you can tell if my solution is ...
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2answers
45 views

How to show that there exists a sequence in $[0,1]$ such that the set of accumulation points of the sequence is $[0,1]$

This is related to homework but I am trying to find a special case first and see if I can generalize it. The problem is to construct some sequence $(x_n)$ in $[0,1]$ such that the accumulation points ...
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0answers
19 views

$\lim\sum_{k=0}^{\lfloor\delta n\rfloor} \frac{n^k}{k!}e^{-n}$ and Poisson distribution

Problem: Let $X_1,X_2\ldots$ be some independent random variables with Poisson distribution with parameter 1. Show that for every $\epsilon > 0$ sequence $S_n-(1-\epsilon)n$ converges to ...
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2answers
31 views

Convergence: infinite series

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}, n\geq\text{some integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
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3answers
52 views

Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always?

I wonder, whether it is always the case $$\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$$ in regards of summation methods for divergent series?
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1answer
28 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
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1answer
43 views

Prove $\{s_n\}$ converges if $\{a_n = s_n + 2s_{n+1}\}$ converges.

Prove that $\{s_n\}$ is convergent if $\{a_n\}$ is convergent where $a_n = s_n + 2s_{n+1}$. This is an old (1950) Putnam question. Clearly $s_n + 2s_{n+1} \rightarrow L$. It looks obvious that $s_n ...
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1answer
30 views

compute very special limit in real number

Let the function $f:$ $\Bbb{R}\to\Bbb{R}$ such that $f(x)=\inf\{|x-me|:m\in\Bbb{Z}\}$ and consider sequence $\{f(n)\}$ then which of the following options is true? a) $\{f(n)\}$ is convergence b)the ...
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2answers
29 views

Convergence of series

Suppose $$\sum_{n=1}^\infty a_n$$converges with $$a_n>0 $$ ,show that $$\sum_{n=1}^\infty \frac{{a_n}^{1/2}}{n} $$ is convergent. Anyone can help me with this? Thanks!,prefer simple method!
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2answers
26 views

Sequence and series : Convergence

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} <= \frac{b_{n+1}}{b_n}, n>=\mathrm{some\space integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
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1answer
81 views

Express $C_n = \cosh(0) + \cosh(1) + \cosh (2) + \dots + \cosh(n)$

Could someone give me some hint of how to do this question please. I've been stuck for more than $3$ hours on this question.
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3answers
65 views

$y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?

My friend and I are currently debating the following question: Let $y_n$ be a sequence in a metric space and assume that the subsequences $y_{2n}$, $y_{2n + 1}$, and $y_{3n}$ all converge. ...
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1answer
35 views

$\sum_{k=1}^\infty \frac {(-1)^{k+1}\cdot 1\cdot 4\cdot 7\cdots(3k-2)}{2^k\cdot k!}$

I'm working on: $$\sum_{k=1}^\infty \frac {(-1)^{k+1}\cdot 1\cdot 4\cdot 7\cdots(3k-2)}{2^k\cdot k!}$$ I've already shown that this series doesn't absolutely converge. I can't use Abel's test ...
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1answer
24 views

How to procede with alternating series

Given some alternating series, the first step is to check whether it's absolutely convergent. Say it's not. Then you use the alternating series test. That test tells you if the series is ...
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1answer
27 views

Define $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n 1-\frac{1}{k^{1+c}}$. Does $\lim_{n\to\infty} a_n=0$ hold? [on hold]

Let $c\in \mathbb{R},c>0$ and define the sequence $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n 1-\frac{1}{k^{1+c}}$. Does $\lim_{n\to\infty} a_n=0$ hold?
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38 views

General term of the sequence $2, 7, 16, 30, 50, 77, 112, 156$ [on hold]

What should be the general term of the sequence $2, 7, 16, 30, 50, 77, 112, 156$?
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1answer
30 views

What sequence of polynomials is equal to $2^n$ for integers $1$ to $k$?

I am trying to prove to someone that no matter how many terms you have of a sequence you can never be 100% sure of the underlying formula. Consider this sequence: $$2^n=1,2,4,8,16,...$$ But just given ...
2
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1answer
23 views

Sum of two divergent sequences with different number of partial limits

Suppose that $(a_n)$ is a sequence which has $1050$ partial limits, and $(b_n)$ is a sequence which has $2750$ partial limits. I'm asked to prove that $(a_n+b_n)$ diverges. So, in general the sum of ...
3
votes
1answer
41 views

Analyzing convergence of series with sine and cosine

Analyze the convergence of the following series: $$\displaystyle\sum\frac{\cos{n}}{\sqrt{n}+\cos{n}}$$ $$\displaystyle\sum\frac{\sin{n}}{\sqrt{n}+\cos{n}}$$ I tried to use the direct comparison test ...
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1answer
35 views

The proof of finding extreme points of the unit ball of $l^1$

Can someone show how to start the proof of finding extreme points of the unit ball of $l^1$? Thanks. Edit: How I've done so far is that Let $B$ be the closed unit ball of $l^1$ Consider any $x_n ...
2
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0answers
32 views

The series may converge, but what about the series / n?

Let $a_i$ be a positive sequence such that $a_i \to 0$. I know that the series $\sum_{i=1}^\infty a_i$ may be divergent. But what about the series divided by $n$; does the following go to 0? ...
0
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1answer
22 views

equivalency of the least upper bound property & convergence of every monotone and bounded sequence in $\mathbb{R}$

I'm aware how to prove convergence of every monotone and bounded sequence in $\mathbb{R}$ by using the completeness of $\mathbb{R}$ (using least upper bound property). But now I want to prove the ...
2
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2answers
36 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
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0answers
24 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...