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

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1answer
42 views

Limiting behaviour of $\sum_{j=0}^\infty|\sum_{k=1}^na_{j+k}|^p$

I would like to prove or disprove the following statement. Suppose that $\{a_j:j\ge0\}$ is an absolutely summable real sequence, i.e. $\sum_{j=0}^\infty|a_j|<\infty$. Then $$ ...
-1
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2answers
43 views

Study the convergence of the series below

Can someone explain to me if this series converges, and what should I test it for in order to study it The series is $$\sum_{n=1}^\infty\frac{1}{2*2^n}$$
3
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1answer
53 views

Taylor series for $\frac{1}{(1+x)^t}$

I'm having some trouble finding the Taylor series for the following function at zero (Maclaurin series). \begin{equation} \frac{1}{(1+x)^t} \end{equation} Where $t$ is a constant that is greater than ...
0
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3answers
104 views

Writing a general formula for an alternating series

I need help for writing the general formula for following alternating series in the form The alternating series are: I feel that 5/(n+8) has something to do with this but I'm not sure how to make ...
3
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0answers
69 views

Prove that a sequence $a_n$ converges if and only if $a_n^3$ converges

I want to prove that $A\ sequence\ a_n\ converges\ \longleftrightarrow\ a_n^3\ converges$ If $a_n$ converges, then by arithmetics, $a_n^3$ converges. Now let $a_n^3$ converge to a real $L$. ...
0
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3answers
31 views

Proving that $\sum\limits_{k = 1}^n k! \cdot k = (n + 1)! - 1$ [duplicate]

Someone may have already asked this question, but I was not able to find it. Prove that $$\sum_{k = 1}^n k! \cdot k = (n + 1)! - 1$$ I tried to use the method that is generally applied to ...
7
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3answers
234 views

Series $\frac{1}{4}+\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}+\cdots$

Find the sum of the series to infinity$$\frac{1}{4}+\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}+\cdots$$ Attempt- I wrote the general term as $$\frac{\binom{2n}{n}}{2^{2n}\cdot ...
0
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1answer
55 views

Series and Absolute Convergence

I was wondering if I could get a hint, and a hidden answer on these two series. We are suppose to find out if they converge absolutely, or conditionally. I am stuck on the test I should use. (1) ...
0
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0answers
34 views

Derive the recursion relation

Consider the nonhomogeneous linear equation $y' = 2y/(1-x) + f(x)$. It is singular at $x=1$, of course, but it is regular at $x=0$ if (the known function) $f(x)$ is analytic there. Assume $y(x) = ...
57
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1answer
674 views

Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
2
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0answers
79 views

Show that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$.

Suppose that $\Sigma_{k=1}^\infty a_k$ converges. Prove that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$. Attemtp: ...
5
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0answers
53 views

Does such a series exist?

This question came up as some puzzle. Does there exist a sequence of real numbers ${c_j}$ such that $\sum{c_j^m} = m$ for all positive integers $m$? I argue no. Suppose there exists such a ...
6
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1answer
118 views

About the series $\sum_{n\geq 0}\frac{1}{(2n+1)^2+k}$ and the digamma function

Let we provide a closed form for $$ S_k = \sum_{n\geq 0}\frac{1}{(2n+1)^2+k} $$ for $k>0$ in terms of elementary functions. It is quite easy to check that $S_k$ can be computed in terms of the ...
-2
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1answer
71 views

Whats the median of $\$4800$ and $\$12000$ [closed]

I'm trying to figure out what the median income would be for a year if I made between $\$4800$ and $\$12000$ in the year.
2
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3answers
171 views

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$ $$\begin{align}\frac{1}{n^4+n^2+1}& =\frac{1}{n^4+2n^2+1-n^2}\\ &=\frac{1}{(n^2+1)^2-n^2}\\ &=\frac{1}{(n^2+n+1)(n^2-n+1)}\\ ...
1
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6answers
190 views

How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$? [duplicate]

How can I prove that $$\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1 \,\,\, ?$$ I do know a way to prove this (see my answer) but I'm curious to know what other approaches could be taken in dealing with it. ...
-1
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3answers
40 views

What is $\lim_{n\rightarrow\infty} \sum_{k=0}^n {n \choose k} (-1)^{k}\frac{1}{k+2}$

Calculate $\lim_{n\rightarrow\infty} \sum_{k=0}^n {n \choose k} (-1)^{k}\frac{1}{k+2}$
1
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2answers
35 views

convergence of a numerical series

I would like to study the convergence of the numerical serie $$ S_n=\sum_{k= 1}^n u_k=\sum_{k= 1}^n \frac{1}{\left(\sqrt[k]{2}+\log k\right)^{k^2}}. $$ I tried the Cauchy rule (i.e. evaluate ...
1
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1answer
53 views

Proving a sequence formula using induction [duplicate]

Suppose for $T_n$: $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}$$ $$T_0=2,\quad T_1=3,\quad T_2=6$$ For integer, $n \ge 3$ I conjectured that: $$T_n = 2^n + n!$$ The above is actually TRUE. Using ...
0
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2answers
292 views

Putnam 1990 A1 Induction Help

A1. $(150,9,1,0,0,0,0,0,1,1,6,33)$ Let $$T_0=2,\quad T_1=3,\quad T_2=6,$$ and for $n\ge3$, $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.$$ The first few terms are ...
4
votes
1answer
31 views

For which real $x$ is this (monster) series convergent?

I'm practicing for an exam, and got to this example: $$\sum_{n=1}^{+\infty} \left (\frac{x^2n^2-2|x|^3n}{1+2xn^2} \right)^{7n}$$ I rearranged the expression to try to check for which $x$ the ...
0
votes
0answers
40 views

Euler type superdivergent

Could you explain where this came from: $$\sum _{k=0}^{\infty } (k!)^2 (-y)^k=\frac{G_{1,3}^{3,1}\left(\frac{1}{y}\mid{{0}\atop{0,0,0}}\right)+2 \left(\log \left(\frac{1}{y}\right)+\log (y)\right) ...
2
votes
1answer
62 views

If $\{{1\over n}\sum_{k=1}^{n}x_k\}_{n=1}^{\infty}$ converges then $\{{1\over n}\sum_{k=1}^{n}{x_k}^2\}_{n=1}^{\infty}$ converges.

Prove or disprove: a. If $\bigg\{{1\over n}\sum_{k=1}^{n}x_k\bigg\}_{n=1}^{\infty}$ converges then $\bigg\{{1\over n}\sum_{k=1}^{n}{x_k}^2\bigg\}_{n=1}^{\infty}$ converges. b. If $\bigg\{{1\over ...
3
votes
1answer
128 views

Inclusion relation between two summability methods

Let $0\leq x<1$ and $s_n$ be a sequence of partial sums of the series $\sum_{n=0}^{\infty}a_n$. It is called that the series $\sum_{n=0}^{\infty}a_n$ is $(A)$ or Abel summable to $s$ if ...
1
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1answer
74 views

Summation of $\tan^{-1}$ series

I am given $$S=\sum\limits_{n=1}^{23}\cot^{-1}\left(1+ \sum\limits_{k=1}^n 2k\right)$$ On expanding the sigma series becomes $$S= 23\cot^{-1}(3)+22\cot^{-1}(5) + \cdots + \cot^{-1}(47)$$ And in tan ...
1
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1answer
26 views

Using the sequential Criterion, give a proof that $\lim\limits_{x\to 0} f(x)$ does not exist, where: $f(x) = -1$, $x \leq 0$ or $x$, $x>0$

Using the sequential Criterion, give a proof that $\lim\limits_{x \to 0} f(x)$ does not exist, where $$f(x) = \begin{cases}-1 & \text{if } x < 0\\ \ \ \ x & \text{if } x \geq 0 ...
1
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2answers
76 views

convergence and divergence of series

I have been working a bit on series and came across two problems I couldn't solve: Determine if the series diverge or converge conditionally/absolutely: 1) $\displaystyle \sum_{n=1}^ \infty ...
1
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1answer
25 views

Sequence converging uniformly on a closed interval

Prove that $x/n→ 0$ uniformly, as $n→ \infty$, on any closed interval $[a,b]$. attempt: Then let $\epsilon > 0$, then there is $N \in N$ such that $n \geq N$ implies $|f_n(x) - f(x)| < ...
1
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1answer
58 views

What number representation is this?

If $p$ is a positive prime number, put $p! = \textrm{lcm}(2, 3, 5, \dots, p) = p\cdots3\cdot2$. Then every non-negative integer can be written uniquely as: $x_1 + 2\cdot x_2 + 3\cdot 2 x_3 + \dots + ...
1
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2answers
84 views

Closed Form for Finite Sum: Product of two Similar Functions

I need to find a closed form expression in terms of $c$, $n$, $x$ and $y$ for $$ \sum_{j=0}^{n}\rho^{c-j}\frac{x^j}{j!}\frac{y^{c-2j}}{\left(c-2j\right)!} $$ where $c$ and $\rho$ are just constants. ...
0
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0answers
61 views

How to prove this g-adic expansion converges?

I am studying series currently, there is a problem set question confusing me which is stated as follows: Let $g\geq 2$ be an integer. Then every real number $0\leq a <1$ can be represented with ...
0
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1answer
22 views

Radius of the convergence of the series

How to calculate the radius of the convergence of the series and what the result will be: $$\sum_{n=0}^{\infty} \left(3^n-2^n+2^{-n} \right) x^n$$ Is it enough to look for $\left( \lim _{n\rightarrow ...
0
votes
2answers
50 views

Show that $\sum_{n=1}^\infty nx^{n-1}$ converges uniformly on $[0,\frac{9}{10}]$

Show that $\sum_{n=1}^\infty nx^{n-1}$ converges uniformly on $[0,\frac{9}{10}]$ First I think we need to show pointwise convergence: $$\sum_{n=1}^\infty nx^{n-1} = \lim_{N\to\infty} ...
0
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1answer
22 views

Find the set of convergence series as $x$

Let $A=\{x>0\mid \sum_{n=1}^{\infty} (\sqrt[n]{x}-1) \text{ is convergence series}\}$. How we can obtain $A$?
2
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1answer
52 views

Find a closed form for the sequence $a_{n+1}= \alpha + \beta a_n^\gamma$

Let $\left(a_k\right)_{k\geq 0}$ s.t. : $$a_{n+1}= \alpha + \beta a_n^\gamma.$$ Is there any closed form ? If not, is there a closed form when $\gamma=-1$ ? ($a_{n+1}=\alpha + \frac{\beta}{a_n}$)
6
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2answers
119 views

How can I prove the $\sum_{n=2}^{\infty }\frac{\log(n)^\frac{1}{k}}{n!}< (e-2)$

How can I prove the $$\sum_{n=2}^{\infty }\frac{\log(n)^\frac{1}{k}}{n!}< (e-2)$$ If $k>1$
0
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1answer
12 views

Sorted sequence algorithm of disctinct integers

I have been been given this task and need some help with it
0
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0answers
21 views

Cauchy's sequence in sup-form

How to prove the below theorem ? Prove that the sequence $f_1$, $f_2$,... converges uniformly to a continuous function, $f\colon [0, 1] \to \mathbb{R}$, by showing that it is a Cauchy sequence in ...
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0answers
52 views

Question about a convergent sequence and inequality

Suppose $0 < x < 1 $ and let $n > 0 $ such that $\frac{1}{n+1} \leq x < \frac{1}{n}$. If there exists a sequence $x_n \downarrow x$, then there exists $N$ such that $\frac{1}{n+1} \leq x_n ...
4
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3answers
50 views

How to solve the summation of series $a^{i}(x+i)$ where $i$ is from $1$ to $N$

I have the following series and I am unable to figure out which series it belongs to and how to solve it $a(x+1)+a^{2}(x+2)+…+a^{N}(x+N)$ Above series is a generalization of my actual series ...
1
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1answer
50 views

prove the limit is greater than lower bound

Question is: $a_n \geq K $ for all $n\in \mathbb{Z^+}$ and converges to $L$. Prove that $L\geq K$. I am thinking of proof by contradiction so I assume that $a_n \geq K $ and $L$ is less than ...
3
votes
1answer
20 views

problem convergent power series expansion such that $f^{(n)}(x)$ and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$

Let $f:(-1,1)$ $\to \mathbb{R}$ such that $f^{(n)}(x)$ exists and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$. Then f has a convergent power series expansion in a neighbourhood ...
1
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1answer
76 views

What does it mean to majorize a quotient?

This is from a proof regarding the uniform and absolute convergence of a power series, from Krantz's Real Analysis and Foundations: We may take the compact subset of $\mathcal{I}$ to be $K = ...
0
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1answer
45 views

Is there a monotone sequence that converges and has a divergent subsequence?

I am trying to figure out an example of a monotone sequence that converges and has a divergent subsequence, but I can't think of one. I know the monotone sequence has to go to 0. Is it even possible ...
0
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1answer
30 views

A little trouble with derivative

I have completed calculus 1 but every so often I still get a litte confused on some basic derivatives. I will give an example, I am not sure if my answer is correct but I will show my work. $ d/dx$ ...
1
vote
1answer
85 views

On the $n^{th}$ day he puts $n$ pennies into the same jar. Which day is the first day on which he has at least $20$ dollars in the jar?

On the first day, Daniel puts one penny into the jar. On the second day he puts $2$ pennies into the same jar. On the $n^{th}$ day he puts $n$ pennies into the same jar. Which day is the first day ...
0
votes
2answers
55 views

Prove $ \lim_{n->\infty}\sqrt[n]{p(n)}=1 $ for polynomial

Let $$ p(x)=\sum_{k=0}^{d}a_kx^k $$ polynomial such that $$ \forall x>0, p(x)>0 $$ Prove that: $$ \lim_{n->\infty}\sqrt[n]{p(n)}=1 $$
2
votes
3answers
58 views

How to prove this sequence doesn't converge to $0$?

Let $b_n$ be a sequence of positive real numbers, and let $c_n=\frac{b_n}{1+b_n}$. Prove that if $b_n$ is unbounded sequence, then $c_{n\:}$ doesn't converege to 0. i thought about this approach: ...
3
votes
1answer
37 views

Eventually constant power tower

Given positive integers $a,m$, let $a_1=a,\ a_{n+1}=a^{a_n}, \forall n\in\mathbb Z_{\ge 1}$. Show that the sequence $(a_n)$ eventually becomes constant $\pmod m$. A solution given is as follows: ...
0
votes
4answers
39 views

Convergence of $\sum_{n=1}^\infty\frac{3}{2n^{p+1}}$

Convergence of $\sum_{n=1}^\infty\frac{3}{2n^{p+1}}$ . What will be the best proof for convergence of this series, which criterion will be the best?