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

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-4
votes
1answer
35 views

What is the pattern of this sequence of numbers?

$5,\,7,\,14,\,16,\,33,\,55,\,82,\,114$ I was given this sequence of numbers, but I can't find the pattern to it. I have put them into different online sequence calculators, but they did not work. ...
0
votes
3answers
26 views

Find some n such that $|s-s_n|< 10^{-3}$

Consider the series $\sum_{n=1}^\infty \frac{1}{n^2}$. Let $s_n$ be the $n$th of the series and $s$ be the sum of the series. Find some $n$ such that$$|s-s_n|< 10^{-3}$$ Can someone please ...
5
votes
1answer
33 views

$p$-series divided by alternating $p$-series = geometric series? Why?

I thought the following equation was interesting: $\dfrac{1 + \frac{1}{2^p} + \frac{1}{3^p} + ... }{1 - \frac{1}{2^p} + \frac{1}{3^p} - ...} = \dfrac{1}{1-2^{1-p}}$ for $p>1$, where $p$ is a real ...
5
votes
5answers
66 views

Why is $f_n(x) = x^n$ not uniformly convergent on $(0, 1)$?

Definition of uniform convergence: For all $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that $d(f_n(x), f(x)) < \epsilon$ for all $n > N \in \mathbb{N}$ and all $x \in (0,1)$. ...
1
vote
1answer
26 views

How to prove $\frac{nx^3}{1 + n^2 x^2}$ converges uniformly on $[1, \infty)$

I know this sequence of functions converges to $0$ pointwise, so I have to show that for all $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $d(\frac{nx^3}{1 + n^2 x^2}) < ...
1
vote
0answers
45 views

How prove that inequality? [on hold]

Let $a_1,a_2,\dotsc,a_n>0$. How prove ...
0
votes
2answers
36 views

Prove these two functions have the same coefficients

We have $\displaystyle p(x) = \frac{1}{1-x}\cdot \frac{1}{1-x^3} \cdot \frac{1}{1-x^5} \cdot \ ...$ and $\displaystyle q(x) = (1+x)\cdot(1+x^2) \cdot (1+x^3)\cdot \ ...$. Let's say that these two ...
0
votes
2answers
53 views

Convergence of $\sum_{n=1}^\infty (2n^{10}+4n^5+1)/(4n^{15}+4n^{12}+5)$

Test whether the following series converges: $$\sum_{n=1}^{\infty}\frac{2\cdot n^{10}+4\cdot n^5+1}{4\cdot n^{15}+4\cdot n^{12}+5}$$ $1.$ I know that it does not make sense to use the root ...
1
vote
1answer
43 views

Calculating the number of terms in arithmetic sequence

I know that if I have a set of numbers, let's say+ $1,2,3,4,5$ I can find the number of terms by subtraction the last term $5$ from the first terms $1$ and then add $1$: $(5-1)+1 = 5$, then the ...
5
votes
2answers
58 views

Find the limit of $\frac{1}{n+1}-\frac{1}{n+2}+…+\frac{(-1)^{n-1}}{2n}$ as $n\to\infty$

Find the limit of $\frac{1}{n+1}-\frac{1}{n+2}+...+\frac{(-1)^{n-1}}{2n}$ as $n\to\infty$. In the previous part of the question I was asked to prove that as $n\to\infty$ ...
2
votes
1answer
43 views

If $d(x_{n+1},x_n)<\frac{1}{n+1}$ then the sequence $\{x_n\}$ is a Cauchy sequence OR not?

Let , $(X,d)$ be a metric space and $\{x_n\}$ be a sequence in $X$. We have, $$d(x_{n+p},x_n)\le d(x_{n+p},x_{n+p-1})+...+d(x_{n+1},x_n)$$ $$\le \frac{1}{n+p}+...+\frac{1}{n+2}+\frac{1}{n+1}\to ...
0
votes
2answers
16 views

How to know the (first term) in arithmetic sequence?

I have an arithmetic sequence and all what I know is the following: The sum of the first 15 terms = 165 The common difference 2 That is: $sum = 165$ $d = 2$ $n = 15$ $a = ?$ I need the ...
1
vote
1answer
60 views

convergency of the sequence $x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+…+(-1)^{n+1}\frac{1}{n}.$

Test the convergency of the sequence $\{x_n\}$ , where $$x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+(-1)^{n+1}\frac{1}{n}.$$ I think the sequence $\{x_n\}$ is convergent. As, ...
-2
votes
3answers
30 views

Determing the radius of convergence of the following power series. [on hold]

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. The series is: $$\sum_{k=0}^\infty ...
3
votes
2answers
36 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ ...
1
vote
1answer
65 views

Is $\sum (\sin n)^n$ diverges? [duplicate]

I saw this series incidentally: $$\sum_{n=1}^\infty (\sin n)^n $$ Result from WolframAlpha seems to say the series diverges but I don't know how to prove it. Thanks for any help!
1
vote
1answer
13 views

Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$

Assume that a sequence $(z_n)$ of complex numbers converges to a nonzero limit. Then Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$ I know I should ...
0
votes
1answer
16 views

When is a series of sums the sum of the series?

In general, if $\Sigma_n (a_n+b_n)$ converges, then it may not be that $\Sigma_n a_n$ and $\Sigma_n b_n$ converge; for example, consider $\Sigma_n (1/n-1/n)$. If instead we know $\Sigma_n a_n$ and ...
0
votes
1answer
40 views

Use series methods to find solution corresponding to..

Use series methods to find solution corresponding to $a_0 = 1$ for the equation $(x+1)y' - y = 0$ Here is my work. Can someone verify that I have the correct solution: So for my final solution I ...
0
votes
0answers
16 views

Calculus scenario involving instantaneous and speed (sequences) [on hold]

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 ...
1
vote
5answers
22 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: $$n(x+2) \over n+1$$
0
votes
2answers
52 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?
1
vote
1answer
25 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 ...
-4
votes
0answers
19 views

Naive Bayes' classifier [on hold]

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?
1
vote
5answers
33 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 ...
1
vote
1answer
22 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 ...
4
votes
2answers
168 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? ...
0
votes
2answers
38 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 $a$?
0
votes
0answers
15 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 ...
3
votes
1answer
27 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 ...
0
votes
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 ...
1
vote
1answer
29 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 ...
0
votes
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.
2
votes
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.
1
vote
0answers
22 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 ...
1
vote
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 ...
1
vote
1answer
20 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 ...
4
votes
3answers
143 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 ...
0
votes
1answer
35 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 ...
0
votes
1answer
33 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 ...
0
votes
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.
3
votes
2answers
55 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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: ...
1
vote
1answer
30 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 ...
4
votes
2answers
46 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 ...
1
vote
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 ...
2
votes
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$ ...
1
vote
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?