Recurrence relations, convergence tests, identifying sequences

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10 views

Series evaluated to $m$ terms, approximating the error

Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shal denote with $E$) when we evaluate it to $m$ terms? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
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1answer
27 views

Partial fraction

How to expand the following expression $\frac{1}{(x^n-1)(x-1)}$ in partial fraction, I think it will be rewritten in terms of geometric series , but how to relate the undefined coefficients ...
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0answers
24 views

partial fraction expansion

I have the following expression $$\frac{\alpha u^2 e^{(\alpha x+y)u}}{(e^{\alpha u}-1)(e^{u}-1)}$$ and I want to expand it in partial fractions $\alpha$ is positive integer
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4answers
34 views

Radius of Convergence of power series of Complex Analysis

I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
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1answer
97 views

Infinite series involving $\sqrt{n}$

I am looking for examples of infinite series, whose sum is expressed as distributions or known functions, with a $\sqrt{n}$ in each term, such as: $$ \sum_{n=0}^{\infty} \sqrt{n} z^n, \quad ...
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0answers
24 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
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2answers
35 views

Sequence convergence of positive numbers

Suppose that $\{a_j\}_j$ is a sequence of real numbers. Suppose for all $j$, $a_j \geq 0$ and the sequence $b_j = \frac{a_j}{1 + a_j}$ converges to $0$. I wish to prove that $a_j$ converges to $0$. ...
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3answers
39 views

Exercise over sequences of real numbers

Let $(a_n)$ be a sequence of nonnegative real numbers such that $$\lim_{n\rightarrow \infty}a_n=0$$ How to prove that exists an infinite number of indices $ n $ such that $a_n\geq a_m$ for all $m\geq ...
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2answers
39 views

Geometric series to calculate price

I decided to add my extension to this question as a new question here. I am trying to represent the following as a geometric series equation: ...
2
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1answer
37 views

what if geometric sequence + geometric sequence

I wrote a program that basicly can find the formula of the sequence that created with any-degree equation. For example if you give my program that sequence: [1926, 2811, 833240, 28778265, 398155842, ...
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1answer
38 views

Sum involving the hypergeometric function, power and factorial functions

I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions. $$ \sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
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1answer
27 views

Difference between convergent sequence and convergent subsequence

I have been thinking about this for a while now. Clearly if a sequence converges then also it will also have a convergent subsequence (take for example the whole sequence). However, I have been told ...
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1answer
17 views

Prove that for n~=n' sum is much smaller than the case with n=n'

Hi I want to prove that this summation is much smaller for $n\neq n'$ than for the case where $n=n'$. I have seen this fact with simulation results. But I don't know how to prove it in mathematics. ...
3
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2answers
90 views

Infinite series question requiring no explanation

Determine if the statement is true or false. No explanation needed. $$\sum_{n=0}^{\infty}\frac{\sin n}{n!}\leq e$$ Although no explanation is needed I was wondering how you would approach this ...
2
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2answers
72 views

Approximating an infinite sum of only odd terms by a definite integral

Consider the infinite Sum $S=\sum\limits_{n\ \text{odd}}^{\infty}\left(\dfrac{1}{nt}\right)^2\left[1-i\left(nt\right)^2\right]^{-1}$ Is there a way to approximate this sum as a contour integral? In ...
2
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2answers
44 views

Prove the following: Product of Roots

$1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} $.... diverges well I don't really know if it does but my gut tells me it does: I can take the log of this product to ...
6
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4answers
115 views

Mathematics Induction

Question: Prove by mathematical induction that $$(1)+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)=\frac{1}{6}n(n+1)(n+2)$$ is true for all positive integers n. Attempt: I did the the induction steps and I ...
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2answers
60 views

Check computation of: $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}ij$

Compute the below sum: $\sum_{i=1}^{n}\sum_{j=1}^{n}ij$ My working: $\sum_{i=1}^{n}\sum_{j=1}^{n}ij = \sum_{i=1}^{n}i\frac{n(n+1)}{2}$ Now since $\frac{n(n+2)}{2}$ is just a constant we can take ...
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0answers
41 views

Prove the inequality $f(m_{1},m)+f(m,m_{2})\ge 1+f(m_{1},m_{2})$

Let $x,y,a_{i}\in R^{n}$ be real numbers,and $0\le m_{1}\le m\le m_{2},0\le a_{i}.i=1,2,\cdots ,k$ and ...
4
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1answer
84 views

Does $\sum \ln\left(\cos\frac{1}{n}\right)$ converge?

I want to know whether the series $\sum \log\left(\cos\frac{1}{n}\right)$ converges or diverges. I have made some attempts to solve this problem, and I work out them here: $\cos(2\theta) = ...
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1answer
21 views

Convergence question of Dirichlet's Test

This is an after-chapter exercise of Dirichlet's Test. Show that if the partial sum $S_n$ of the series $\displaystyle\sum_{k=1}^{\infty} a_k \leq Mn^r$, for some $r<1$, then the series ...
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2answers
41 views

Prove that the series is non-absolutely convergent.

$$a_n = \int_{(2n-2)\pi}^{(2n-1)\pi} \dfrac{\sin t}{t} dt$$ The series is $$\sum_{n=1}^{\infty}a_n$$ I tried using the Cauchy criterion, and this let me with the next inequality: $$\left| S_m - S_n ...
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1answer
44 views

show the result is right when $f\in C(\Bbb{R})$, but when $f$ is only Riemann integrable. Is it right?

Assume $f(x) \in C(\Bbb{R})$, and $$S_n(x)=\sum_{k=1}^{n}\frac{1}{n}f\left(x+\frac{k}{n}\right),n=1,2,\cdots,$$ show that: $\forall [a,b] \subset \Bbb{R}$ , $S_n$ converges uniformly and if $f(x)$ ...
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1answer
44 views

What is the half-derivative of zeta at $s=0$ (and how to compute it)?

I'm trying to understand the concept of fractional derivatives and am fiddling with the examples at wikipedia. The a'th derivative of a monomial in x, where a can be fractional is accordingly $$ {d^a ...
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1answer
60 views

For which $x$ values this series $\sum_{n=1}^{\infty}\frac 1n \cos^2(nx)$ is convergent.

I have tried using Dirichlet's Test with $\cos^2nx = \dfrac{1+\cos(2nx)}{2} = \frac 12 + \dfrac{\cos(2nx)}{2}$, this show that I can't bound the partial sums. What another test may I apply?
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2answers
58 views

Series, Looks Simple, but I am Stuck

I promised a friend that I could help her about math questions. Yet, I am stuck with a series question. I have written the open form of each term. And I have split the general term into multiples. I ...
1
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1answer
45 views

nth term test for divergence - help

$$\sum_{n=1}^{\infty} \left(\dfrac{n}{n+1}\right)^n$$ to show that this diverges should I use the $n^{th}$ term test? So far I have substituted infinity for $n$. Could I use L'hopital's rule to ...
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1answer
32 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
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0answers
32 views

Solve For a Generating Function

Special Functions: Legendre, Bessel, Elliptic, and Confluent Hypergeometric Functions Considering the following functions of $f_{n}(x)$ defined by: $$ (a)---- ...
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1answer
33 views

question about convergence series

I'm not sure I understand why $$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}(4pq)^n<\infty $$ when $$p\ne0.5 , p+q=1$$ I know that $\sum_{i=1}^{\infty}\frac{1}{\sqrt{n}}=\infty$ But why when we have ...
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1answer
25 views

Formula for working out an ID number by given set of coordinates

I'm designing an online game and having a bit of a mental block coding the navigation system. It's designed on a 2 dimensional grid, each cell has an ID 0...n, n being the total number of cells in the ...
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2answers
55 views

ratio test and divergence

I need to be able to prove that this series converges. I know I need to use the ratio test but I do not know how to go about doing it. Any help is much appreciated! thank you
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1answer
132 views

Find a closed form for this sequence

$$a_1 = 1; a_2 = 9; a_{n+2} = \frac{a_{n+1}a_n}{6a_n - 9a_{n+1}}$$ I need to find non-recurring formula for $a_n$. Is there any good way to do this? The only one comes to mind is to guess the formula ...
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1answer
61 views

Let $a_k=\dfrac{x^k}{k!}$ Show that $\dfrac{a_{k+1}}{a_k}\leq\dfrac12$

Fix $x\geq0$ and let $a_k=\dfrac{x^k}{k!}$ Show that $\dfrac{a_{k+1}}{a_k}\leq\dfrac12$ for sufficiently large k, say $k\geq N$
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1answer
64 views

integration as limit of a sum

If $f$ is continuous on $[0, 1]$ then $$\lim_ {n\to\infty}\sum_{j=0}^{\lfloor n/2\rfloor} \frac1{n}f\left(\frac {j}{n}\right) = ? $$ will the answer be that the limit exists and is equal to $ ...
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2answers
90 views

How to calculate the sum of $(n-1)^2+(n-2)^2+…+1$? [closed]

How to calculate the sum of the following series? $$(n-1)^2+(n-2)^2+...+1$$Thank you in advance
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5answers
327 views

Finding the smallest integer $n$ such that $1+1/2+1/3+…+1/n≥9$?

I am trying to find the smallest $n$ such that $1+1/2+1/3+....+1/n \geq 9$, I wrote a program in C++ and found that the smallest $n$ is $4550$. Is there any mathematical method to solve this ...
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5answers
320 views
+50

limit of : $a_{n+2} =\frac{1}{a_n} + \frac{1}{a_{n+1}}$

$a_n$ is a real sequence, $a_1,a_2$ are positive and for all $n>2$ : $$ a_{n+2} =\frac{1}{a_{n+1}} + \frac{1}{a_{n}}.$$ Prove that: $\displaystyle \lim_{n\to \infty} a_n$ exists, then find it. ...
0
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0answers
46 views

Show that if $\sum_{n=1}^{\infty}{a_n}$ converges, then $\lim_{n \rightarrow \infty}{na_n}=0$ [duplicate]

Show that if $\sum_{n=1}^{\infty}{a_n}$ converges, then $\lim_{n \rightarrow \infty}{na_n}=0$ Suppose $\lim_{n \rightarrow \infty}{na_n} \neq0$. Since $\sum_{n=1}^{\infty}{a_n}$ converges, by ...
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1answer
41 views

Struggling to understand a couple of concepts with series

I have two questions: neither of which are homework problems but certainly pertain to my ability to do the homework. The first regards the harmonic series. The question has been answered often here ...
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1answer
59 views

Convergence of the series $\sum a_n$ implies the convergence of $\sum \frac{\sqrt a_n}{n}$, if $a_n>0$ [duplicate]

I need help to solve following problem from Rudin's Mathematical analysis book: Convergence of the series $\sum a_n$ implies the convergence of $\sum \dfrac{\sqrt {a_n}}{n}$, if $a_n>0$ I ...
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1answer
62 views

Suppose $f_n(x)=x^n - x^{2n}$ , $x \in [0,1]$. Dose the sequence of functions $\lbrace f_n \rbrace$ converge uniformly?

Suppose $f_n(x)=x^n - x^{2n}$ , $x \in [0,1]$. Dose the sequence of functions $\lbrace f_n \rbrace$ converge uniformly? I try to show that the sequence is Cauchy. But I get stuck. Suppose $n>m$. ...
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2answers
64 views

Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, ...
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2answers
55 views

Suppose $\sum_{n=1}^{\infty}{x_n} < \infty$,$\sum_{n=1}^{\infty}{|y_n - y_{n+1}|} < \infty$

Suppose $\sum_{n=1}^{\infty}{x_n} < \infty$,$\sum_{n=1}^{\infty}{|y_n - y_{n+1}|} < \infty$. Then prove that $\sum_{n=1}^{\infty}{x_ny_n}$ converges. Since $\sum_{n=1}^{\infty}{x_n} < ...
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1answer
81 views

Proof that Riemann-integrability is preserved by changing the function at a converging sequence of points

Let $(x_n)^{\infty}_{n=1}\in[a,b]$ such that $\lim\limits_{n\to\infty}x_n=l$. Let $g:[a,b]\to \mathbb{R}$ be bounded function. Assume that there exists a Riemann integrable function $f:[a,b]\to ...
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1answer
59 views

(When) Is $\sum\limits_{k=0}^\infty(-1)^k(a+bk)^Nx^k$ positive for all $0 < x < 1$?

I have been stuck with this problem for while. So hope someone will give me a hint how to solve it. For $a$ and $b$ are positive real number, is it true that $$ S = \sum\limits_{k = 0}^\infty ...
14
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5answers
238 views

Convergent or Divergent? $\sum_{n=1}^\infty\left(2^{\frac1{n}}-1\right)$

Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent? $$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$ I can't think of anything to compare it against. The integral looks too hard: ...
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2answers
61 views

Finding the closed form for a sequence

My teacher isn't great with explaining his work and the book we have doesn't cover anything like this. He wants us to find a closed form for the sequence defined by: $P_{0} = 0$ $P_{1} = 1$ ...
2
votes
2answers
34 views

limit inequality for bounded sequences

Let $(a_n)$, $(b_n)$ be bounded sequences. Prove that if $a_n + b_n \rightarrow c$, then $c\le \limsup \,a_n + \liminf\, b_n$. I have tried a proof. I don't know if it is correct. Proof: since ...
3
votes
2answers
76 views

Continuous function and increasing property

Assume that $f(x)$ is continuous function. Also, we have that for all $y>0$ there exists $\epsilon>0$ such that for all $x \in (y-\epsilon, y)$ we have $f(x)<f(y)$. From this I want to prove ...

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