For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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3
votes
2answers
61 views

function approximation using series

I was trying to approximate a function by another one using some kind of a series. Let $$ f(x) = m\cdot \left(1-\sqrt{\frac x2\biggm/\left(1+\frac x2\right)}\right) $$ I'm trying to approximate ...
5
votes
4answers
101 views

$\sum_{n=1}^{\infty} \frac{n^2}{ n!}$ equals [duplicate]

$ \sum_{n=1}^{\infty} \frac{n^2}{ n!} $ equals I'm not able to convert in any standard series? Any hints?
0
votes
1answer
74 views

For small x, Taylor Series to determine constants n and C in the approximation

cos(x) - e^(-(x^2)/2) I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst ...
3
votes
1answer
52 views

Summation approaches zero

if I have a sequence $A=$ $\frac{1}{n} \sum _{k=1}^n kC_k$ and $nC_n$ approaches $0$ as $n$ approaches infinity, then how can I show that $A$ goes to $0$? I just need it for my proof of the Tauber's ...
3
votes
2answers
30 views

Problem on compactness of convergent sequence in a metric space

Let $\{x_n\} \to x$ be a convergent sequence in a metric space. I want to prove that the set $\{x_n\}\cup x$ is compact. I guess it follows easily from the fact that any seqence in $\Bbb R$ has a ...
1
vote
3answers
66 views

For what value of $\alpha$ does this series converge?

The Question is for what value of $\alpha$, does the series $$\sum_{n=0}^{\infty} (\sqrt{n^\alpha + 1} - \sqrt{n^\alpha})$$ By setting $\alpha \leq 0$, we can show the series diverges by vanishing ...
1
vote
3answers
127 views

What is the point of the term “monotonic” when analyzing sequences?

If my understanding is good, a sequence is monotonic if it increase or decrease. In my homework, I was asked to indicate if some sequences were increasing or decreasing, and also to specify if it was ...
2
votes
2answers
42 views

I'm unsure which test to use for this series, and how to prove?

I want to determine if this series converges or diverges: $$\sum\limits_{n=1}^\infty{\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}}$$ I tried the Ratio Test at first, and didn't get anywhere with that. I'm ...
2
votes
2answers
47 views

Finding the sum of $\sum_{n=0}^{\infty}\frac {(x+1)^{n+2}}{3^n}$

Find the sum of the series and for which values of $x$ does it converge: $$\sum_{n=0}^{\infty}\frac {(x+1)^{n+2}}{3^n} $$ My attempt: ...
0
votes
0answers
60 views

Proof of Neumann Series

Consider the theorem Given that $\lambda_k$ is the $k$th eigenvalue of $A$, the following identity holds when $|\lambda_k | < 1$: $$ (I + A)^{-1} = \sum_{i = 1}^{\infty} (-1)^iA^i. $$ Is ...
0
votes
2answers
52 views

Convergence of the sequnce and series

so I'm preparing for my calculus exam and I have no ideea how to solve this sequence and also compute the series. here is the function I have tried to conjugate, to force the common factor out, and ...
1
vote
3answers
64 views

Summation to infinity: $\sum_{r = n + 1}^\infty \frac{1}{r(r + 1)}$

How would I get an answer for this, I know it's a telescopic series but I'm not sure how you can sum to infinity? I have managed to get a formula in terms of $n$ as $1 + \left(\frac{1}{r(r + ...
1
vote
1answer
37 views

Convergence domain of: $\sum^{\infty}_{n=1}\frac {(3x-5)^{2n-3}n!}{n^n}$

Find the convergence domain of: $\displaystyle\sum^{\infty}_{n=1}\frac {(3x-5)^{2n-3}n!}{n^n}$ My attempt: Define $t=3x-5$ so: $\displaystyle\sum^{\infty}_{n=1}\frac {t^{2n-3}n!}{n^n}$. ...
1
vote
1answer
46 views

How to show this sequence convereges, and calculate its lim?

Let $\:x\in \mathbb{R}$. How to calculate the lim of:$$\lim _{n\to \infty }\frac{\left[1x\right]+\left[2x\right]+\left[3x\right]+...+\left[nx\right]}{n^2}$$ ([x] is the floor function of x). So far ...
1
vote
3answers
34 views

How to prove the limit of $\frac{r^n}{n^{-5}}$ ($0<r<1$) is 0

Intuitively, I know it's true. $r^n$ goes to 0 exponentially fast, which should be much faster than $n^{-5}$ when $n$ becomes large. But I want to prove it. Please help me.
3
votes
4answers
99 views

Infinite sequence series. Limit

If $0<x<1$ and $$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+\ldots+\frac{x^{2^n}}{1-x^{2^{n+1}}},$$ then $\lim_{n\to\infty} A_n$ is $$\text{a) }\ \dfrac{x}{1-x} \qquad\qquad \text{b) }\ ...
2
votes
1answer
53 views

Find $a_{n}$ if $(a_{n})$ is a sequence such that $a_{1} := 1$ and $\frac{1}{a_{n+1}} = \frac{2}{a_{n}} + 3$ for $n \geq 2$?

This problem is weird. By the initial condition $a_{1} = 1$ we have $a_{2} = \frac{1}{5}$ and so on. But is there really a pattern for $a_{n}$? I guess this problem is that kind of problems that ...
1
vote
2answers
48 views

Find an integer $k$ such that $a_{k} = 2^{261}$?

Let $a_{1} := 2$ and $$a_{k} := \frac{2^{(k+1)(k+2)/2}}{\prod\limits_{j=1}^{k-1}a_{j}}$$ for all integers $k \geq 2.$ The problem is to find an integer $k$ such that $a_{k} = 2^{261}.$ The ...
7
votes
2answers
103 views

$\sum_{i=1}^{89} \sin^{2n} (\frac{\pi}{180}i)$ is a dyadic rational

Last year's Euclid contest had a problem asking for the rational value of $\sum_{i=1}^{89} \sin^{6} (\frac{\pi}{180}i)$. I tested this sum for different even powers, and the result was always a ...
0
votes
4answers
94 views

Convergence of $\sum\limits_{n=2}^\infty \ln\left(\dfrac{n^2}{n^2-1}\right)$

I made sure it passed the nth term test. Next I thought the easiest way, given that it's wrapped in ln, would be to use log rules to make it $\ln(n^2)-\ln(n^2-1)$ and then compare it to $\dfrac ...
1
vote
0answers
59 views

Truncated Binomial Series

Can the truncated binomial series be expressed as a closed form \begin{align} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{k} x^{k} \end{align}
16
votes
2answers
340 views

Do any of these sequences have infinitely-many distinct iterates under run-length substitution?

Let $$S = \{x \in \{1,2\}^\mathbb{N}: \ \text{every run in }x\text{ has finite length}\}$$ and define $$T: S\to \mathbb{N}^\mathbb{N} $$ such that for any $x\in S$, ${T}x$ is the sequence of ...
0
votes
3answers
40 views

Can someone help me with this Comparison Test problem involving cos?

I need to determine if this series converges: $$\sum{\frac{cos^2n}{n^2}}$$ So far, I know I'm supposed to use the Comparison Test. And I compare it with the series $\sum{\frac{1}{n^2}}$, which we ...
1
vote
3answers
83 views

Find the sum of the series $1-\ln(2) +\frac{\ln(2)^2}{2!}-\frac{\ln(2)^3}{3!}+\cdots$

I can represent it as an infinite sum in the form: $\sum_{n=0}^\infty \frac{(-1)^n\ln(2)^n}{n!}$, and I can use various series tests to prove it converges, but I'm unsure how to find the actual sum.
1
vote
0answers
50 views

Evaluate $\sum\limits_{i=0}^{n}\frac{1}{6^i}$

I'm asked to evaluate: $$\sum\limits_{i=0}^{n}\frac{1}{6^i}$$ Since it's a geometric series, I got $\dfrac{1-\frac{1}{6^n}}{\frac56}$ but I think it could be wrong.
0
votes
1answer
47 views

Summation to Infinity

Could someone explain a step by step solution to this? It's 1 mark but I missed the lesson and can't find anything on summation to infinity or how you'd go about it. It's basic but my teacher ...
1
vote
1answer
72 views

Fourier Series Expansion, error in coefficients?

After reworking the problem many times I keep getting the same (incorrect?) answer. So the problem as stated is Find the Fourier expansion of : $$ f(x) = \begin{cases} x &\text{ if }0 ...
3
votes
1answer
44 views

I'm looking for the sequence $\Big\{ \frac{n}{n+1}\sin{\frac{n\pi}{2}} \Big\}$ converges or diverges.

I'm looking for the sequence $$\Big\{ \frac{n}{n+1}\sin{\frac{n\pi}{2}} \Big\}$$ converges or diverges. I tried ...
0
votes
2answers
43 views

Convergence/divergence of $\sum_{n=1}^\infty (-1)^n \frac {\ln (n)} {\sqrt n}$

$$\sum_{n=1}^\infty (-1)^n \frac {\ln (n)} {\sqrt n}$$ How can I determine/show whether the above series converges or diverges? It seems like I should use the Alternating Series Test, but I don't ...
0
votes
0answers
49 views

Technical term for summation of sequential numbers

Given $n$, I wish to do the following operation: $f(n) = n + (n-1) + (n-2) + ... + 1$. For example, $f(4) = 10$. What is the technical term for this kind of operation?
1
vote
3answers
53 views

Interchange of sum and limit in sequence algebra

As you may know, Let $ {a_n} $ and $ {b_n} $ be convergent sequence with limit L, M respectively, then the following is true $ \lim_{n\to \infty} (a_n + b_n) = \lim_{n\to \infty} a_n +\lim_{n\to ...
0
votes
1answer
33 views

how to check the convergence of the following series

I was having a hard time grasping the concept of convergence and how to check convergence for the following series: $$ a_n= \frac{1}{\sqrt{n^2+n}} ...
2
votes
1answer
57 views

Asymptotic for $\sum a_nb_n$ if asymptotic for $\sum a_n, \sum b_n$ are known

Let us assume that $a_n>0$ and $b_n>0$ for each n. Also let $$ \sum_{n\leq x} a_n \sim f(x) $$ and $$ \sum_{n\leq x} b_n \sim g(x) $$. What can we say about the asymptotic on $\sum_{n \leq x} ...
0
votes
3answers
55 views

Convergence of $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{1/n}}$

Does the series $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{1/n}}$ converge? converge in absolute value or conditionally? It's easy to see that in absolute value the general term tends ...
3
votes
1answer
89 views

What is the sum of this infinite series

So the question is - $\displaystyle S = \sum_{n=1}^\infty{\frac{1}{10^n}\left(\begin{matrix}2n\\ n\end{matrix}\right)}$. Find $S$. I tried converting the $n^{th}$ term as a difference of two terms ...
0
votes
3answers
132 views

Does $\sum_{n=2}^ \infty \frac 1 {n \sqrt {ln \ n}}$ converge?

I want to figure out if this sum converges or diverges: $$\sum_{n=2}^ \infty \frac 1 {n \sqrt {ln \ n}}$$ I tried comparing it to the harmonic series, but this is less than that so it was no use. The ...
2
votes
1answer
93 views

The Banach–Mazur distance for finite-dimensional $\ell_p$

Let $\ell_p$ denote the usual infinite-dimensional sequence space, and if $n\in\mathbb{Z}^+$ then we let $\ell_p^n$ denote its $n$-dimensional counterpart. Conjecture. Let $1\leq p<\infty$. ...
2
votes
0answers
31 views

Wondering about limit comparison test and its application

I have a question about the limit comparison test. In particular if it is valid to use in the question I will post. I am interested in determining wether the series $$\sum_{n=2}^\infty ...
0
votes
3answers
55 views

Prove that $\sum\limits_{i=1}^{n^2}2i=n^4+n^2$ [closed]

My teacher has given us this problem to do during the weekend but me and my friends could not do by any means: $$\sum\limits_{i=1}^{n^2}2i=n^4+n^2$$
2
votes
2answers
90 views

Approximating the value of Euler's constant?

I'm asked the following: Using the series that defines $\gamma$, Euler's constant, what's the minimum number of terms that we have to sum in order to calculate $\gamma$ with an error less than $2 ...
3
votes
3answers
90 views

prove: $\lim_{n \rightarrow \infty} x^{1/n} = 1$

I have to prove that $\lim_{n \rightarrow \infty}$ $x^{1/n} = 1$ for $x > 0$. I splitted it up in 3 cases: $x = 1:$ $1^{1/n} = 1$ $\forall$ $n$, so $\lim_{n \rightarrow \infty}$ $x^{1/n} = 1$ if ...
0
votes
3answers
76 views

Need to evaluate the series $\sum_{i = 0}^n i^5$ [duplicate]

I need help in evaluating the following sum: $$\sum_{i = 0}^n i^5 $$ I can evaluate series when they are arithmetic or geometric but I don't know how to solve this one.
0
votes
1answer
33 views

Sigma Series Sumate To Infinity

How do I summate to infinity? I know I need to give the equation in terms of "n" and split it up into two different summations and subtract them but I don't know how you can physically get a value up ...
0
votes
2answers
58 views

$K$ is a limit point of $(x_{n})_{n = 0}^\infty \Rightarrow$ There is a subsequence of $(x_{n})_{n = 0}^\infty$ that converges to $K$.

I have to prove this exercise for my math study. It's form Terence Tao's Analysis I: Assume $(x_{n})_{n = 0}^\infty$ is a sequence of real numbers, and $K \in \mathbb{R}$. I have to prove the ...
3
votes
1answer
140 views

Limit points and subsequences

I'm having trouble with proving this exercise: Let $(a_{n})_{n = 0}^\infty$ be a sequence of real numbers, and $L \in \mathbb{R}$ . Prove: $L$ is a limit point of $(a_{n})_{n = 0}^\infty ...
1
vote
1answer
124 views

Sandwich rule for sequences

I'm taking a first year calculus course and I'm stuck on a concept with sequences. The question asks to use the sandwich rule to find the following limits of sequences: (1) and (2) I've shown ...
3
votes
1answer
238 views

Checking convergence of $\sum\frac{\sin nx}{n}$

Consider the sequence $$f_n(x)=\sum_{k=1}^{n}\frac{\sin kx}{k}\quad x\in \mathbb{R}$$ now we have to check convergence of $\{f_n\}$. Now, I used Dirichlet's criterion to show that $f_n$ converges ...
1
vote
1answer
57 views

Radius of convergence of a power series whose coeffecients are “discontinuous”

I have a power series: $s(x)=\sum_0^\infty a_n x^n$ with $a_n= \begin{cases} 1, & \text{if $n$ is a square number} \\ 0, & \text{otherwise} \end{cases}$ What is the radius of convergence ...
6
votes
1answer
354 views

Does $\sum_{n=1}^{\infty}\ln\left(n\sin\left(\frac{1}{n}\right)\right)$ converge?

I must determine whether the following series converges: $$\sum_{n=1}^{\infty}\ln\left(n\sin\left(\frac{1}{n}\right)\right)$$ I know that in general, I must use the limit comparison test, but I ...
0
votes
1answer
37 views

Does the $n$th term test work for integrals?

I know of several convergence/ divergence tests for infinite sums, but strangely the only tests I know for improper integrals (which seemingly are just continuous versions of infinite sums) are the ...