Tagged Questions

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

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4
votes
0answers
11 views

Evaluating $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$

I would appreciate to understand the main steps giving the evaluation of this series: $$ S=\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$$ where $H_n$ is the harmonic number. I've tried with no ...
1
vote
0answers
16 views

How much freedom do we have when working with sequences?

Consider the sequence $$a_n = \sum_{k = 1}^n \frac{1}{k} - \log(n)$$ and suppose we are to show it is Cauchy. (Do not solve the exercise for me. Spoilers are not welcome.) While there maybe are better ...
1
vote
1answer
18 views

Uniform Convergence of an infinite series

Show that for each $r> 0$, $\sum_{k=0}^{\infty} \frac{1}{k^2 -z}$ converges uniformly on the set $E_r = \{z: |z| \leq r, z\neq k^2$ for $k = 0,1,2,3, ...\}$. I tried to use the Weierstrass M-test ...
2
votes
1answer
44 views

A Series Problem from Calculus-2 course

This problem is from calculus-2 course. The basic knowledge includes integral test and $p$-series test. Find an $N$ so that $$\sum_{n=1}^\infty {1\over n^4}$$ is between $$\sum_{n=1}^N {1\over ...
0
votes
1answer
30 views

What about the convergence of the geometric mean sequence of the terms of a given convergent sequence?

Let $(a_n)$ be a sequence of positive real numbers such that $$ \lim_{n\to} a_n = a.$$ Let $b_n \colon= \sqrt[n]{a_1 \cdot a_2 \cdot a_3 \cdots a_n}$ and $c_n \colon= a_n^{a_n}$. Then what can we say ...
1
vote
0answers
23 views

Finding a Counter Example - Limits of integrals of an increasing sequence of Borel measurable functions

I need to find a counter example to the following problem. I'm trying to think of some, but maybe I'm not creative. I'm not sure. Let $h$ and $h_1, h_2, h_3, ...$ be Borel measurable functions such ...
0
votes
0answers
25 views

Showing a sequence of integrals converges.

I'm having trouble with this problem - I don't even know how to begin. Thoughts? Solutions with explanation? Please help! Let $f$ be a bounded continuous function on $\mathbb{R}$. Prove that $$ ...
0
votes
2answers
48 views

How to calculate this limit?

Let $$a_n \colon= \frac{1}{\sqrt[n]{n!}}$$ for $n = 1, 2, 3, \ldots$. Then how to decide about the convergence or otherwise of the sequence $(a_n)$? And if this sequence IS convergent, then how to ...
1
vote
2answers
37 views

How is $\sqrt{2}$, for example, in the closure of $\mathbb{Q}$ in the usual metric space $\mathbb{R}$?

Let $\mathbb{R}$ be the set of all real numbers under the usual metric $d$ defined as follows: $$d(x,y) \colon= |x-y|$$ for all $x$, $y$ in $\mathbb{R}$, and let $\mathbb{Q}$ be the set of all ...
1
vote
1answer
17 views

A sequence of Lebesgue integrable functions.

My friend and I came upon a problem in Real Analysis. It called for a sequence of Lebesgue integrable functions $(f_n)$ converging everywhere to a Lebesgue integrable function $f$ such that $$ ...
0
votes
0answers
18 views

Help find radius of convergence for $\Sigma_{n=0}^{\infty}a^{n^2}z^n$ [on hold]

I need to find the radius of convergence for the power series $\Sigma_{n=0}^{\infty}a^{n^2}z^n$, where $a > 0$ and $z \in \mathbb{C}$. Any help?
1
vote
2answers
36 views

Help showing $\sum_{n=2}^{\infty}(\log n)^{-\log n}$ converges.

I need to show that $\sum_{n=2}^{\infty}(\log n)^{-\log n}$ converges without using the integral test. Any help?
1
vote
2answers
19 views

Display cans of food in a square-based pyramid.

Full question: The manger of another grocery store asks a stock clerk to arrange a display of canned vegetables in a square-based pyramid (top is one can, 4 cans under then, 9 cans under top 2 ...
1
vote
2answers
25 views

sum of series using method of difference

Please I have a problem with finding d sum of the sequence 3x4 ,4x5 ,5x6,...... using method of difference ....most books I use only explain partial fractions, but I have found the $n$th term to be ...
1
vote
1answer
20 views

convergence of a series..

This might be ridiculously easy but I just forgot about series. Consider the series $\sum_{k=1}^\infty \frac{1}{k^2-2}$. Does it converge? What about $\sum_{k=1}^\infty \frac{1}{k^2-r}$ for any ...
0
votes
0answers
18 views

Differentiation for Taylor series expansion

Knowing $f'(x)=x f(x)+1$ for any $x$ and $f(0)=1$, find the Maclaurin series ($x=0$) to the cubic term. Process: Differentiate $f'(x)$ twice So, the expansion is $f(x)=1+x+x/2+x/6+...$ Would ...
0
votes
2answers
55 views

Help with Baby Rudin Theorem 3.29

Theorem 3.29 If $p >1$ then $\sum_{n=2}^{\infty} \frac{1}{n(\log n)^p}$ converges; if $p \leq 1$, the series diverges. Proof: The monotonicity of the logarithmic function implies that $\{\log ...
0
votes
0answers
23 views

Does $A193201$ count the partitions of $n$ of arbitrary dimension?

By my count, these sequences match for $n=0\ldots6$, where partitions that are the same after relabeling dimensions are considered equivalent (i.e., the dimensions are unordered). For example, for ...
4
votes
0answers
28 views

Question regarding an inequality

How to prove that $$ \frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+x_1^2+\cdots+x_n^2}<\sqrt{n} $$ knowing that $(x_n)$ is a positive sequence ? I looked up all kinds of ...
1
vote
2answers
25 views

Question about Mobius function.

Let $N \in \mathbb{N}.$ I would know if is it true that $$-\underset{k\mid N}{\sum}\mu\left(k\right)\log\left(k\right)>0.$$I know that $$-\mu\left(k\right)\log\left(k\right)=\underset{r\mid ...
0
votes
4answers
58 views

How to show that $\sum_{n=1}^{\infty}(-1)^n \sin(a/n)$ converges but not absolutely.

How to show that, for any fixed constant $a\in(0,1)$, the series $$\sum_{n=1}^{\infty}(-1)^n \sin\frac{a}{n}$$ is convergent yet not absolutely convergent. My idea is to express sin(x) as series but ...
2
votes
0answers
20 views

Asymptotics of inverse Laplace transform of a function with an essential singularity?

Let $h$ be the function $$ h(x) = \sum_{k\geq0} \frac{(ix)^k}{k!}\zeta(2k), $$ with the Laplace transform $$ \tilde h(s) = -\frac{\pi}{2s}\sqrt{i/s}\cot\left(\pi\sqrt{i/s}\right), $$ which has an ...
0
votes
2answers
20 views

On converging and diverging sequences and their respective arithmetic mean

I'm working on a problem set which was given by our analysis lecturer (a) Let $(a_n)$ be a convergent sequence with limit $a$. show that the arithmetic mean $$s_n := \frac{1}{n} \sum_{k=1}^n ...
0
votes
1answer
31 views

Show that there is a sequence ${a_n}$ such that for every real number x, there is a subsequence of ${a_n}$ converging to x

I came across this question in a textbook with no solutions and I'm having trouble with where to start. I'm thinking that somewhere I might need to use the Bolzano-Weierstrass Theorem but I am unsure ...
1
vote
2answers
63 views

Question regarding convergent series

If the series with general term $a_n^2$ converges, why does the series with general term $a_n/n$ converge as well??? A peer of mine showed me this, but I really don't find it obvious and I really ...
2
votes
2answers
89 views

Finding the infinite sum of $e^{-n}$ using integrals

I am trying to understand this: $\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though: $= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$ $= \displaystyle ...
6
votes
2answers
64 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
14 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} ...
4
votes
1answer
73 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
35 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
66 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 ...
1
vote
2answers
35 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
26 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
15 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
52 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
69 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
85 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
14 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
17 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
50 views

If series converges? (By comparison test)

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
59 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
51 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
26 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
36 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
39 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
22 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
42 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
14 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} = ...