For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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20 views

Finding a formula for a kth element in a sequence

I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$ The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - ...
0
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1answer
21 views

Unable to choose functions for evaluating a limit using the Squeeze Theorem

Evaluate $$\lim_{n\to \infty}\dfrac{1}{\sqrt{n^2}}+\dfrac{1}{\sqrt{n^2+1}}+...+\dfrac{1}{\sqrt{n^2+2n}}$$ $$$$ I'm supposed to solve this problem using the Squeeze Theorem. I had selected the ...
4
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0answers
39 views

Show there are only a finite number of integers with $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)} $ an integer

Show there are only a finite number of integral $(a_i)_{i=1}^n$ such that $2\le a_i \le a_{i+1}$ and $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)} $ is an integer. My question is inspired by ...
1
vote
1answer
22 views

Is $Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$ a compact operator?

Is the operator $A$ defined by $$Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$$ a compact operator? It only has finitely non-zero dimensions, so does this mean ...
-1
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0answers
21 views

Laurent Series, How it is done

Suppose that a series $$\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$ converges to analytic function $X(z)$ in some annulus $R_1<|z|<R_2$. That sum $X(z)$ is called the z-transform of $x[n]$ $(n=0,\mp ...
0
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1answer
26 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
0
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1answer
16 views

Taylor series integration

I am having trouble with the following question: Integrate the Taylor series $$e^{(-t^2)} = \sum^\infty_{n=0} \frac{(-t^2)^n}{n!}$$ term-by-term to obtain the Taylor series for erf (error function) ...
0
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1answer
17 views

Prove $-\frac{\ln(1-x^2)}{1-x^2}=H_1x^2+H_2x^4+H_3x^6+H_4x^8+\cdots$

$H_n$ is nth the harmonic numbers $x<1$ (1) $$-\frac{\ln(1-x^2)}{1-x^2}=H_1x^2+H_2x^4+H_3x^6+H_4x^8+\cdots$$ A different approach of representing $\ln(x)$ let expand out the series ...
2
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2answers
72 views

Complicated series converges to $\pi$.

How do I get this result? $$\frac {426880 \sqrt {10005}}{\large \sum_{k = 0}^{\infty}\frac {(6k)!(545140134k + 13591409)}{(k!)^3 (3k)! (-640320)^{3k}}} = \pi$$ It seems formidable. Context: I came ...
4
votes
2answers
43 views

For which values of real $\alpha, \beta$ does $\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$ converge?

I was wondering how does the series $$\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$$ behave for real $\alpha, \beta > 0$. My approach: firstly I considered the case $\alpha = \beta > 2$. ...
4
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2answers
30 views

Show that the sequence $x_n=\Big[\frac32x_{n+1}\Big]$, $x_1=2$ contains an infinite set of odd numbers and an infinite set of even numbers.

I am having a hard time proving this Let $x_n$ be a sequence such that $x_{n+1}=\Big[\frac32x_n\Big]$ for $n\gt1$, where $[x]$ denotes the nearest integer function and $x_1=2$. Show that ...
2
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3answers
78 views

Series convergence $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$

Series $A = \sum_{n=1}^\infty\frac{1}{\ln(n+1/n)}$ diverges by the comparison test (wolfram). I want to compare $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$ with series $A$. How can I prove that ...
1
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0answers
9 views

Maclaurin series and sereis converge [duplicate]

$f(x)$ is a continuos fuction in $[0,1]$ and differentiable twice on $0$. $U_n=(-1)^{n}f(\frac{1}{n})$ I need to prove that: $1.$ if $f(0)=0$ then $\sum_{n=1}^\infty U_n$ converge. $2.$ if ...
0
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3answers
19 views

If $f_n:X\to\mathbb{R}$ converge uniformly a $f$, then there is $N\in\mathbb{N}$ such that $n>N\to f_n(X)\subseteq U$.

Let $X$ be a compact, $U$ open set and $f:X\to\mathbb{R}$ continuous such that $f(X)\subseteq U$. If $f_n:X\to\mathbb{R}$ converge uniformly a $f$, then there is $N\in\mathbb{N}$ such that $n>N$ ...
1
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0answers
48 views

$\zeta(1)=\frac12\ln(2)$? Did I do something wrong?

I attempted to calculate $\zeta(1)$ and I got $\frac12\ln(2)$. $$\zeta(1)=\lim_{\epsilon\to0}\frac{\zeta(1+\epsilon)+\zeta(1-\epsilon)}2$$ ...
1
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3answers
65 views

Limit of $\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}$ when $n\to\infty$

I have to show the convergence of the series $$\lim\limits_{n \to \infty}a_n=\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}.$$ I am quite sure that the limit is 1.5. I wanted to show this ...
1
vote
2answers
39 views

$(f_n) $ converge uniformly on $X$.

Let $(f_n)$ be a sequence function continuous $f_n:X\to \mathbb{R}$ that converge uniformly on $D\subseteq X$ dense set. Then $(f_n) $ converge uniformly on $X$. can someone help me with this. If X ...
1
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4answers
108 views

calculating $\lim_{n\to\infty} \frac{1+\frac{1}{2}+…+\frac{1}{n}}{n}$

How can I prove that $\lim_{n\to\infty} \frac{1+\frac{1}{2}+...+\frac{1}{n}}{n}=0$? I can't use $1+\frac{1}{2}+\cdots +\frac{1}{n}\approx \log n$ I've tried to use the following: $\lim_{n\to\infty} ...
2
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1answer
57 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = ...
1
vote
1answer
35 views

Approximate integral using Taylor Series

I have to approximate this integral with an error lesser than 0.1 using Taylor Series. This is the integral: $$\int_0^1 \arctan(\frac{1}{x^{10}}) dx$$ If I understood, I have to determinate the Taylor ...
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0answers
51 views

How to integrate ${\int_0^{\pi} \frac{\sin (x)}{A-B\sin^3( x)} dx}$ [on hold]

$${2\over\int_0^{\pi} \frac{\sin x}{A-B\sin^3 x} dx} =\text{?}$$
1
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0answers
14 views

Order Size estimation of converging sum used for approximation of logarithm

I know it can be shown that $\log n=\sum_{i=1}^\infty \frac{(n-1)^i}{in^i}$ for $\forall n\in\Re$ where $n\ge1$ For given natural m, I tried to find the order size of k = f(m,n) in order for the ...
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0answers
27 views

Summation of Series [on hold]

The question is that you have to find $\sum_{r=1}^n rx^{1/r}$ I tried various things such taking log of the general term , differentiating but couldn't reach anywhere .
2
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1answer
21 views

Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin ...
1
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0answers
28 views

Radius of convergence of the series $\sum\limits_{-\infty}^{\infty}(2^{-n}+4^{-n}) z^n$

I'm trying to find for what values of $z\in\mathbb{C}$ the series $$\sum_{n=-\infty}^{\infty}(2^{-n}+4^{-n})z^n$$ converges. My main methods are the nth root test and the ratio test. I believe it can ...
0
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1answer
21 views

Solving differential equations using series

Suppose $v \in \mathbb{R}$ and $$y\left(x\right)=x^v\left(1+\sum^{\infty}_{n=1}a_nx^n\right)$$ where the series converges for any $x \in \left(-r,r\right), r>0$ If $y\left(x\right)$ is a solution ...
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0answers
20 views

Proof this limit superior is finite.

Let $\{ w_n \}$ be a sequence of non-negative numbers and put $M_n=\sum_{k=1}^n w_k^2 \xrightarrow{n\to\infty} \infty $. Proof that $$\limsup_{n\to\infty} \dfrac{\ln \ln \sqrt{M_n \ln \ln M_n} }{\ln ...
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0answers
26 views

When finding Laurent series when to use partial fractions?

When finding the Laurent series of $$f(z):=\frac{1}{z(z-1)(z-2)}$$ valid in the region $1<|z-2|<2$ for example do we just use partial fractions to break $f(z)$ up and the just find the Laurent ...
0
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1answer
43 views

Intuition behind proving that $\lim_{n \to \infty} \frac{1}{b_n} = \frac{1}{b}$

Okay, so I'm having a bit of an issue understanding Rosenlicht's proof that $$\lim_{n \to \infty} \frac{1}{b_n} = \frac{1}{b}$$ I've worked out the proof and I have written it down myself, but ...
0
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3answers
29 views

Proving a subsequence doesn't converge

When I want to prove that a sequence doesn't converge by showing that it's subsequence doesn't converge , can i use the limit comparison test? (Usually used for series) . for example - $$ \sum_{n ...
0
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1answer
33 views

Properties of Bessel Functions

I have shown that the Bessel function $$J_n\left(x\right)=\dfrac{x^n}{2^n}\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^kx^{2k}}{2^{2k}k!\left(n+k\right)!}$$ satisfies the property ...
3
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1answer
18 views

Theorem 3.44 in Baby Rudin: Can we replace the coefficients with their absolute values?

Here's Theorem 3.44 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Suppose the radius of convergence of $\sum c_n z^n $ is $1$, and suppose $c_0 \geq c_1 \geq c_2 ...
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0answers
17 views

What is the sum of $n$ terms of numbers common in two A.P.? [on hold]

What is the sum of the first hundred numbers common to the two arithmetic progressions: $$17,21, \ldots$$ $$16,21, \ldots$$
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0answers
25 views

Riemann’s Rearrangement Theorem

I have a question regarding to th Riemann’s Rearrangement Theorem: I want to prove that if $\sum a_n$ is conditionally convergent then there exist a series with the same terms but with different order ...
1
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0answers
10 views

Examples 3.35 (a) and (b) in Baby Rudin: Limit Superior and limit inferior of a couple of sequences

This question is related to Examples 3.35 (a) and (b) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition, p. 67. Let us consider the series $$ \frac 1 2 + \frac 1 3 + ...
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1answer
16 views

General term of the sequence and series

Whats is the general term of the sequence? for each value of n corresponding to the value of $f(n)$ $ f(n)=2,\;2,\;2,\;4,\;4,\;4,\;4,\;6,\;6,\;6,\;6,\;8,\;8,\;8,\;8,\;10,\;10,\;10,\;10 $; where $n = ...
4
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3answers
73 views

What is $\lim_{x\to \infty} 2\sqrt{x}- \sum_{n=1}^x {1\over \sqrt{n}}$? [duplicate]

I ask this because I noticed the partial sum $\sum_{n=1}^x {1\over \sqrt{n}}$ is very close to $2\sqrt{x}$, so close in fact that it appears their difference approaches a constant value, like $H_x$ ...
3
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1answer
40 views

summation of $\sum_{k=0}^{\infty}x^{n^{k}}$

Let $x\in (0,1)$ and $n\in 2\mathbb{N}+1$ be fixed. the series $$\sum_{k=0}^{\infty}{x^{n^{k}}}$$ is convergent by Ratio Test. what is the sum of the series ?
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3answers
96 views

Is there an infinite sequence of real numbers $a_1, a_2, a_3,… $ such that ${a_1}^m+{a_2}^m+a_3^m+…=m$ for every positive integer $m$?

Is there an infinite sequence of real numbers $a_1, a_2, a_3,...$ such that ${a_1}^m+{a_2}^m+a_3^m+...=m$ for every positive integer $m$? I tried assuming that the sequence $a_1^m, a_2^m,...$ ...
2
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1answer
34 views

Which exponents r>0 is the limit finite

I am trying to find values of $r>0$ such that $\lim\limits_{n\rightarrow \infty} \sum\limits_{k=1}^{n^2}\frac{n^{r-1}}{n^r+k^r}$ is finite. I have tried to use integral methods for this limit such ...
0
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0answers
20 views

summation of $\sum_{k=0}^{\infty}{q^{\sum_{i=0}^{k}{n^{i}}}}$

Let $q\in (0,1)$ be fixed. Consider the sequence $\{q^{\sum_{i=0}^{k}{n^{i}}}\}_{k=0}^{\infty}$, where $n$ is a fixed odd positive integer. This sequence is convergent to zero by dini's theorem. set ...
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0answers
27 views

Power Quantum Series And It's Sum.

Let $I$ be the interval $(-\theta, \theta), \theta=q^{\frac{1}{1- n}}$, $n\in 2\mathbb{N}+1$ and $q\in (0,1)$ are fixed. Define a function $h(t):=qt^{n}$. One can see that the $k$-th order iteration ...
2
votes
1answer
40 views

How to find the Laurent series of $\frac{1}{z^4(1-z)^2}$ for |z|>1?

A hint is given that $$\frac{1}{(1-\frac{1}{z})^2} = \frac{z^2}{(1-z)^2}$$ and we know that $$\frac{1}{1-w} = \sum_{n=0}^{\infty} w^n$$ for $|w|<1$. I don't know how to make ...
0
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0answers
9 views

Is the derivative product rule true for Bochner spaces?

If $u\in L^{2}(0,T; L^{2}(\Sigma))$ with $u_{t}\in L^{2}(0,T; H^{-1}(\Sigma))$ and $v\in L^{2}(0,T; H^{1}_{0}(\Sigma))$ with $v_{t}\in L^{2}(0,T; L^{2}(\Sigma))$ is it true that $$ ...
1
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2answers
72 views

Finding a closed form for $\sum_{k=1}^{\infty}(-1)^{n-1}\frac{\zeta(2n+1)}{2^{2n+1}}$

I've been playing with series involving odd values of the zeta function. Some time ago I found the following closed form $$ \sum_{k=1}^{\infty}\frac{\eta(2k+1)}{2^{2k+1}}=\frac{1}{2}-\ln(2) $$ and ...
1
vote
1answer
12 views

Does convergence of bounded, increasing sequences generalize from the set of real numbers to arbitrary partially ordered sets?

If $t$ is any real number, then I can find a strictly increasing sequence $(t_n)$ of real numbers converging to $t$. E.g. $t_n = t - \frac{1}{n}$ would do. Does this generalize to arbitrary partially ...
0
votes
1answer
28 views

Double sequences and iterated limits

Let a double sequence be defined as a function from $\mathbb{N} \times \mathbb{N} \to \mathbb{R}$, which we write as $(a_{m, n})$. We say that $(a_{m, n})$ converges to $a$, in symbols $(a_{m,n}) \to ...
1
vote
2answers
46 views

Prove or disprove the series converge conditionally

I have the following series - $$ \sum_{n = 1}^\infty {\frac {(-1)^{k(n)}}{n}} $$ when $$k(n)=\begin{cases} 1 &; \quad n \ \text{=$3m$, for $m$ natural number}\\ 2 &; \quad n \ ...
1
vote
0answers
31 views

the sum-int exchange

Why is the sum-int exchange allowed in the following equality ($f$ is a $C^1$ function): $$ \sum_{m\geq 1} \int_0^{\pi}f'(t) \frac{\sin((2m+1)t)}{2m+1} dt = \int_0^{\pi}f'(t) \sum_{m\geq ...
0
votes
1answer
36 views

Convergence of $\sum_{k \geq 1} e^{-tk} \cos kz$

I would like to find the convergence of the series $\sum_{k \geq 1} e^{-tk} \cos kz$. Clearly, this series converge in using the comparison test or the integral. How could I get an explicit function ...