Tagged Questions

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

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0answers
8 views

Maclaurin's series

Q1, What conditions must be met for Maclaurin's series to be valid? Q2, Using Maclaurin's series, the power series for cos4t as far as the term in t to the power 6 is? Please show workings as this ...
3
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1answer
18 views

Find an $N$ so that $\sum_{n=1}^\infty {\ln n\over n^2}$ is between $\sum_{n=1}^N {\ln n\over n^2}$ and $\sum_{n=1}^N {\ln n\over n^2} + 0.005$

This is an exercise from Calculus-2 course "Sequence and Series": Find an $N$ so that $$\sum_{n=1}^\infty {\ln n\over n^2}$$ is between $$\sum_{n=1}^N {\ln n\over n^2}$$ and $$\sum_{n=1}^N {\ln ...
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1answer
36 views

How can we prove that it is an increasing sequence? [duplicate]

We know that the limit of this sequence $\left(1+\frac1x\right)^x$ is $e$. But how can I prove that it is an increasing sequence?
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1answer
21 views

Sum of the series of numbers in an array

You are given a number $x$ and an array A[1,2....n] storing $n$ positive numbers such that $$A[1]+A[2]+....+A[i]≤A[i+1], \space \forall i<1.$$ Design a polynomial time algorithm to determine if ...
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0answers
53 views

Sums in $\mathbb N^3$

Assume that $a_{n,m,k}\geq 0$ is a sequence, $n,m,k\in \mathbb N$, such that $$\sum_{n,m,k\in \mathbb N} a_{n,m,k}^2 <\infty$$ i.e. it is in $\ell^2 (\mathbb N^3)$. I want to prove the following: ...
3
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1answer
21 views

Is absolute convergence the same as unconditional convergence?

The Riemann Series Theorem states that a conditionally convergent series, upon permutation, can be made to converge to any value, or diverge. I want to know when we can guarantee that a series ...
4
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2answers
43 views

Solutions to $\sum{a_n} = \prod{(a_n+1)}$

$$\sum_n^\infty{a_n} = \prod_n^\infty{(a_n+1)}$$ Can you give a nontrivial example of a real sequence which satisfies this equation? By "trivial" I mean sequences such as $-1,1,0,0,0\dots$ which ...
3
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1answer
46 views

Does this simple sum converge

I'm trying to determine whether the sum $$S=\frac{2}{1}+\frac{2\cdot 5}{1\cdot 5}+\frac{2\cdot 5\cdot 8}{1\cdot 5\cdot 9}+...+\frac{2\cdot 5\cdot 8...(3n-1)}{1\cdot 5\cdot 9...(4n-3)}+...$$ converges ...
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2answers
37 views

Proving Cauchy sequences

Let $x_n$ be a sequence in the real number set satisfying $|x_{n+2}-x_{n+1}|\le r|x_{n+1}-x_n|$ for all $n\in\Bbb R$, where $0<r<1$. Let $\{x_n\}_{n\in\Bbb N}\subset\Bbb R$ be a sequence ...
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1answer
35 views

prove the limit using definition. $\lim_{(n) \rightarrow (\infty)} \frac{7n^2+9n-17}{4n^2-5n+6}= 7/4$

$\lim_{(n) \rightarrow (\infty)} \frac{7n^2+9n-17}{4n^2-5n+6}= 7/4$ I got that $|\frac{71n-110}{16n^2-20n+24}| < $ $\epsilon$ , how do I continue from here ?
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4answers
92 views

Calculate $\sum_{n=1}^{\infty}(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4})$

I am trying to calculate the following series: $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)}$$ and I managed to reduce it to this term ...
1
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2answers
23 views

Sequence in Sequence Challenge

Suppose that there are two arithmetic sequences $a_n$ and $b_n$. Given that ${a_b}_{20}$ + ${b_a}_{14} = 2014$. What is the value of ${a_b}_{14}$ + ${b_a}_{20}$
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1answer
63 views

Can we pull out a constant of a divergent series?

I know that if a series converges, the following applies: $$ \sum_{n=i}^\infty c a_n = c \sum_{n=i}^\infty a_n $$ However, I can't seem to find any info on whether this holds for diverging series as ...
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3answers
35 views

prove the limit using definition.

1) $\lim_{(n) \rightarrow (\infty)} \frac{4n+7}{2n-4}= 2$ 2) $\lim_{(n) \rightarrow (\infty)} \frac{7n^2+9n-17}{4n^2-5n+6}= 7/4$ in first question I got that $ \frac{15}{|2n-4|}$ < $\epsilon$ , ...
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3answers
70 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 ...
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0answers
24 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
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1answer
19 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
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1answer
52 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 ...
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1answer
32 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 ...
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1answer
29 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 ...
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0answers
28 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 $$ ...
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3answers
63 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 ...
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2answers
39 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
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1answer
20 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
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0answers
20 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?
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2answers
38 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
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2answers
20 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 ...
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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
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1answer
21 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 ...
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0answers
21 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 ...
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2answers
58 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 ...
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0answers
37 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 ...
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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 ...
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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
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4answers
64 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
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0answers
21 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 ...
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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
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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 ...
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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
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2answers
93 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
66 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
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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
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1answer
77 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
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1answer
36 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 $1_10_20_3...0_n$, $0_11_20_3...0_n$, ... , ...
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5answers
68 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
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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.
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0answers
26 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
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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 ...