Tagged Questions

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

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5
votes
4answers
48 views

Find $S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+…+\frac{2n-1}{2^n}+…$

I'm trying to calculate $S$ where $$S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+...+\frac{2n-1}{2^n}+...$$ I know that the answer is $3$, and I also know "the idea" of how to get to the ...
0
votes
1answer
12 views

How do I see formaly that $\liminf_{n \rightarrow \infty} |f_n|^p = |f|^p$ for $p \in [1, \infty)$?

Let $(f_n)_{n \in \mathbb N}$ be a sequence of functions s.t $\lim_{n \rightarrow \infty} f_n = f$, where $f_n : X \rightarrow \mathbb R$. How do I see formaly that $\liminf_{n \rightarrow \infty} ...
3
votes
0answers
32 views

How to prove $ \sum_{r=1}^{k-1} \binom{k}{r}\cdot r^r \cdot (k-r)^{k-r-1} = k^k-k^{k-1} $

How to prove the following identity: $$ \sum_{r=1}^{k-1} \binom{k}{r}\cdot r^r \cdot (k-r)^{k-r-1} = k^k-k^{k-1} $$ I have no idea how to tackle it because of the $r^r$. Any help is highly ...
1
vote
1answer
32 views

function from a Binary sequence to the Natural Numbers

I apologize if this is a duplicate question. I don't know enough terminology to thoroughly search. However, given a sequence of binary numbers $1_10_20_3...0_n$, $0_11_20_3...0_n$, ... , ...
0
votes
5answers
62 views

Convergence or Divergence of $\left \{\frac{n!}{n^n} \right\} $

Determine whether the sequence is convergent or divergent. If it is divergent, find its limit. $$ \left\{\frac{n!}{n^n} \right\} $$ I tried to write out some of the terms of this sequence, and ...
0
votes
2answers
31 views

What is the pattern for this sequence?

I know that it increments by 1 until the (10n + 1)th term, where it increments by the term #. I don't know how to represent this entire pattern as an equation or summation of some sort.
-3
votes
0answers
25 views

How to write recurrence relations from a verbal description(Question from Oxford math admission test)? [on hold]

Questions are interesting because they only require primary math skill. They have general patterns that developing the problem from specific to general. The 5th and 7th questions in the paper(linked) ...
0
votes
1answer
23 views

How to prove that this series is positive

For each $s\in\{z\in\mathbb{C}:\operatorname{Re} s> 0\}$, let $$F(s) = \sum_{n\geq 1}\frac{(-1)^{n+1}}{n^s}.$$ How to prove that, for each $0 < s <1$, $F(s) > 0.$
0
votes
1answer
14 views

Find a recursive definition for the sequences

The first sequence given is 3, 7, 16, 41, 77,.... I really am quite stuck on this because I can't seem to find any relationship between one term and the terms prior to it. I first noticed that it ...
3
votes
3answers
50 views

find common ratio of $\sum_{k=1}^\infty \frac{1}{k(k+1)}$

I have this problem, I need to find the sum. $$\sum_{k=1}^\infty \frac{1}{k(k+1)} = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\cdots+\frac{1}{k(k+1)}$$ The problem is that the ...
1
vote
0answers
28 views

Does a specific function exist with these properties?

lets say i have a function where: $[p,c,n,k] \in \mathbb{Z}$ defined some way in this function $ f(x) = \sum_{q=0}^{n} \sum_{v=1}^{k-q+1}\frac{c(k+1-v)!(p-k)!}{(k+1-v-q)!q!(p-k-q)!}x^{k-v+1-q} = ...
2
votes
2answers
60 views

Is there a formula for $k\pi ^n$, if $n$ is an odd number and $k$ is a rational number?

I always find the formula for $k\pi ^n$ when $n$ is an even number and $k$ is a rational number, but I did't find for an odd number.
3
votes
3answers
82 views

$\frac{1}{1+2}+\frac{1}{1+2+3}+\dots+\frac{1}{1+2+3+\dots+x}=\frac{2011}{2013}$

I want to see OTHER approaches than this one. Make sure they are significantly different and not a direct restatement. ...
1
vote
1answer
12 views

Check whether the following expressions are equivalent

$$(x-1)(x^{n+1}-1)=(x^2-2x+1)(x^n+x^{n-1}+\cdots+x+1)$$ Are the following expressions listed above equivalent? If they are, how to show that?
0
votes
0answers
16 views

bounded and convergent sub sequences

We are given with a bounded sequence $x_n$ and let $$ y_k = \sup_{n\ge k} x_n= \sup\{x_k,x_{k+1},….\}. $$ How will we prove that sequence $y_k$ is decreasing and bdd?
2
votes
3answers
40 views

If series converges?

For what value of real constant $a$ does the following series converge? $$ 1+(1/\sqrt{3})-(a/\sqrt{2})+(1/\sqrt{5})+(1/\sqrt{7})-(a/\sqrt{4})+(1/\sqrt{9})+(1/\sqrt{11})-(a/\sqrt{6})... $$ I do not ...
3
votes
0answers
55 views

Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
3
votes
0answers
42 views

What is the most elementary way of proving a sequence is free of non-trivial squares?

Given the sequence A001921 $$ 0, 7, 104, 1455, 20272, 282359, 3932760, 54776287, 762935264, 10626317415, 148005508552, 2061450802319, 28712305723920, 399910829332567, \dots $$ which obeys the ...
1
vote
2answers
23 views

What function does this trigonometric series represent?

I wonder what known function does this trigonometric series represent? $$ f(x)=\sum_{k=1}^\infty \frac{1}{k^{3/2}}\sin{(kx)} $$ with $$ -\pi\le x\le \pi $$ and $$ f(x)=f(x+2\pi) $$ And how is it ...
0
votes
1answer
35 views

How do we find the $n+2$th term of the series $1 + (3+5) + (7+9+11)+\dots$?

We have the series $1 + (3+5) + (7+9+11)+\dots$. We need to find the $n+2$th term and hence summation of the series up to this term. However hard we try we do not seem to be able to fit this ...
1
vote
1answer
36 views

clarification on convergence of series?

I get the idea if a sequence is convergent then $$|b_n-L|<\epsilon$$ for n>=N. but I did not get it with series convergence. $$|\sum_{k=m}^{n}a_k|<\epsilon$$ shouldn't it be ...
1
vote
2answers
20 views

How to determine if this converges?

I find this one to be hard, I tried expanding $\cos$ into a Taylor series, but that didn't work out well because I couldn't apply $p$-series... $$ \sum^{\infty}_{n=1} n (1-\cos(\pi/n)) $$
1
vote
3answers
36 views

How to determine if this series converges?

Does this series converge? I tried using limit comparison, and I don't know what to try next... $$\sum^{\infty}_{n=1}(1-\cos (\pi/n)) $$
0
votes
1answer
12 views

Show that zero sequences satisfy the following equation

I am working on the following problem and got puzzled: $\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = ...
1
vote
3answers
32 views

Convergence and Divergence of this series

Does the series $\dfrac{n\ln(n)+4}{n^2}$ converge or diverges? Which test should be applied? I've tried integral test but I couldn't figure out.
2
votes
2answers
53 views

Finding the $nth$ partial sum for $e^{-n}$

Here is the question: $$\displaystyle \sum_{n=1}^{\infty} e^{-n}$$ Instead of using the formula of $\large\frac{1}{1-r}$ I want to try to get the partial sums. $S_1 = e^{-1}$ $S_2 = e^{-1} + ...
4
votes
2answers
13 views

Control ratio of geometric series through its sum

A geometric series $S_n$ is the sum of the $n$ first elements of a geometric sequence $u_n$: $$u_n = ar^n \space \forall n \in \mathbb{N}^*$$ with $u_0$ defined, and: $$S_n = \sum_{k = 0}^{k = n - ...
0
votes
1answer
32 views

the sequence $a_n$ $\rightarrow$ $0$ [on hold]

Prove that, if a bounded sequence $(a_n)$ has only subsequential limit $0$ then $a_n$ $\rightarrow$ $0$. Give me some hint!
1
vote
1answer
20 views

convergence and nested logs

The problem is to test convergence for the series: $\sum^\infty_{n=3}1/(\ln n)^{\ln(\ln(n))}$ I tried manipulating the log term (by means of ...
5
votes
3answers
54 views

Divergence of $a_{n+1}=\sqrt{2a_n+3}$?

I am wondering what I am missing from my proof. I would like to show that the limit of the sequence $$a_{n+1}=\sqrt{2a_n+3},\,\, a_1=4,$$ goes to $\infty$, as $n \rightarrow \infty$. Is there ...
3
votes
3answers
132 views

does this sequence necessarily converge?

Let $\{x_n\}$ be a real sequence that satisfies $|x_{n+1} - x_n| < \frac{1}{n}$ for all $n \geq 1$. Suppose we know that $\{x_n\}$ is bounded, then must $\{x_n\}$ converge?
3
votes
2answers
29 views

Test for convergence for $\ln \frac{n^2}{n^2-1}$

I've tried to figure out if this converges using the comparison test, and the ratio test, but with no luck: $\sum^\infty_{n=2} \ln(n^2/(n^2-1))$. I'd appreciate any help
1
vote
2answers
36 views

Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$.

Let $S$ be a bounded set. Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$. Note: If $\sup S$ is in $S$, it’s sufficient to define $s_n = \sup S$ for all ...
0
votes
1answer
22 views

Applying the monotone convergence theorem

Recently learned about the monotone convergence theorem. I have the sequence: $x_n = \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}$ I need help proving that it is increasing and bounded, ...
1
vote
1answer
17 views

Monotonically decreasing sequences

Suppose $(a_n),(b_n)$ be positive sequences such that $(a_n)$ decreases to $0$, monotonically. If lim$_{n\rightarrow \infty}\frac{b_n}{a_n}=1$, does it imply that $(b_n)$ decreases to $0$ ...
5
votes
5answers
59 views

If $x_n \to \pm \infty$ and $x_n y_n$ converges, prove that $y_n \to 0$

Let $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ be sequences in $\mathbb{R}$ such that $x_n \to \pm \infty$ and $(x_ny_n)_{n \in \mathbb{N}}$ converges. Show that $y_n \to 0$. This ...
1
vote
2answers
45 views

Convergence of a series with positive terms

Let $(a_n)_n$ be a strictly positive sequence . How to prove that the series $$ \sum\limits_{n = 1}^\infty {\frac{{a_n }}{{(a_1 + \cdots + a_n )^2 }}} $$ converges ? Any ideas ?
0
votes
1answer
17 views

Divegent series test

I have this infinite series. What test would you use to show that it is divergent? I tried limit, ratio and tried to compare it with $2^{1/n}$ $\sum_{n=1}^\infty {2^{1/n}}-1$ Thanks
0
votes
2answers
15 views

Application of Alternating series test

Is the series $\displaystyle\sum_{n=1}^\infty \frac{(-1)^n n^n}{n!e^n}$ convergent? This is inconclusive by ratio test. I tried to use root test, but ended up with ...
0
votes
0answers
10 views

Convergence in distribution of a serie

How could we prove that this serie converge in distribution to a centered gaussian variable ? $$ \frac{1}{\sqrt{n^3}} \sum_{i,j,k = 1}^{n} x_{i,j} x_{j,k} x_{k,i} $$ with for all $ i,j \in ...
2
votes
0answers
13 views

Rearrangement of absolutely convergent series

I would be very grateful if someone would verify whether my proof below is correct. Many thanks. Theorem. $\,$ Let $(b_k)$ be a rearrangement of the complex sequence $(a_k)$. If $\sum_{k\geq 0}a_k = ...
3
votes
0answers
43 views

How many numbers are products of $p^p$?

Consider the set $\mathcal{S}=\{1,4,16,27,\ldots\}$ of numbers which are products of numbers of the form $p^p$ for $p$ prime. ($\mathcal{S}$ is A072873 in the OEIS.) Note that multiple primes are ...
0
votes
1answer
36 views

Difference between power series method and Frobenius method

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$. Then ...
5
votes
0answers
68 views

Finding the closed form of $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k n}$

A while ago I computed pretty easily the series $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k+ n}$ and then I thought of tackling the case where we have the product instead of ...
0
votes
1answer
23 views

Evaluating a limit that involves a summation

I was solving a physics problem and I got this expression: $E=\lim_{N \to \infty}\left[\dfrac{k_0Q}{2R^2}\dfrac{1}{N}\sum\limits_{i=0}^{N/2-1}\left(\sec{\dfrac{i\pi}{N}}\right)\right]$ I'm not sure ...
2
votes
4answers
100 views

Find the Sum of the Series: $1/(x+1) + 2/(x^2 + 1) + 4/(x^4 +1) +\cdots$ $n$ terms

Find the sum of $n$ terms the following series: $$\frac1{x+1} + \frac2{x^2 + 1} + \frac4{x^4 +1} +\cdots\qquad n\text{ terms}$$ $t_n$ seems to be $\dfrac{2^{n-1}}{x^{2^{n-1}} + 1}$ But after that ...
2
votes
3answers
54 views

Is the following series converging $\sum_{n=2}\dfrac{1}{(\ln n)^{\ln n}}$ [duplicate]

Is the following series converging $\sum_{n=2}\dfrac{1}{(\ln n)^{\ln n}}$ I am not able to compare this with anything, can some show the way
0
votes
2answers
42 views

Show that $\sum a_n$ diverges if $\sum \log (\tfrac{1}{1-a_n})$ diverges

Let $\{a_n\}$ be a sequence that satisfy $0\le a_n<1$ for all $n$. Given that the series $\displaystyle \sum_{n=1}^{\infty} \log \left(\frac1{1-a_n}\right)$ diverges. Prove or disprove ...
0
votes
1answer
26 views

prove $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+…+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$

Let $x_1,x_2,x_3$,... be a sequence of nonnegative real numbers. Prove that $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+...+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$ I ...
1
vote
3answers
54 views

e as sum of an infinite series

I read that $e = \sum_{i=0}^\infty$$ 1\over n!$. This isn't immediately obvious to me, and I can't find proof of this. Can somebody explain to me, how do I prove this from definition $e = \lim_{n\to ...