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

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2
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1answer
54 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
50 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
1answer
81 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
37 views

Showing a sequence of integrals converges. [duplicate]

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
3answers
114 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
134 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
37 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 $$ ...
1
vote
2answers
80 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
49 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
28 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 ...
2
votes
2answers
110 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 ...
7
votes
0answers
170 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 ...
7
votes
2answers
110 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
43 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
91 views

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

How to show that, for any fixed constant $a\in(0,1)$, the series $$ \sum_{n=1}^{\infty}(-1)^n \sin\left(\frac{a}{n}\right). $$ is convergent yet not absolutely convergent. My idea is to express ...
2
votes
0answers
50 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
39 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
39 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
95 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
129 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
74 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
16 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
110 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
63 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
81 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
53 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.
0
votes
1answer
28 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
2answers
34 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
63 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
32 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} = ...
1
vote
2answers
79 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
91 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
15 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?
2
votes
3answers
61 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 ...
6
votes
1answer
100 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 ...
4
votes
3answers
262 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
36 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 ...
1
vote
1answer
65 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
55 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
28 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
58 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... $$\sum^{\infty}_{n=1}(1-\cos (\pi/n)) $$
1
vote
1answer
19 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} = ...
0
votes
3answers
39 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
64 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} + ...
5
votes
2answers
26 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 - ...
1
vote
1answer
26 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
67 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
137 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
39 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
87 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 ...