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

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1
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2answers
59 views

Sum of the series $\sum \frac{n}{2^{n}}$

I know that the series converges by d'Alembert ratio test, where $lim\left ( \frac{A_{n+1}}{A_{n}} \right )= \frac{1}{2}$, but I don't know how to calculate the sum of the serie. Thanks for the help.
5
votes
5answers
103 views

Showing $\displaystyle\lim_{z\rightarrow ^{-}1}\displaystyle\sum_{n\geq 0}z^{2^n}$ does not exist

I have been trying to bound this below, as the TA suggested, by some taylor series of a function I know diverges at $x=1$, like $\log(\frac{1}{1-x})$ taylor expanded around zero: ...
9
votes
1answer
65 views

Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows: $p_0 = a_0$ $p_1 = a_0x+a_1$ $p_2 = a_0x^2+a_1x+a_2$ $p_n = ...
2
votes
2answers
61 views

sum of a function series

what would be the first step to determine sum of $\sum\limits_{n=1}^{\infty}ne^{-n^2/4x}.$ I think I should try putting $y=e^{-1/4x}$. Then $y$ changes from $0$ to $1$ and I get ...
2
votes
1answer
43 views

Maximum value of $S_n(x)=\frac{4}{\pi} \sum_{k=1}^n \frac{\sin(2k-1)x}{2k-1}$

I'm doing the exercise $11.19$ from Apostol Real Analysis: Let $S_n(x)=\frac{4}{\pi} \sum_{k=1}^n \frac{\sin((2k-1)x)}{2k-1}$. Prove that $S_n(\frac{\pi}{2n}) \geq S_n(\frac{m \pi}{2n})$ for ...
1
vote
2answers
91 views

Edwards Differential Calculus for Beginners, 1896. Chapter 4, Question 82

This question asks for an evaluation of an infinite series assuming only knowledge of basic differential calculus. I couldn't figure it out, but user 'Dr. MV' gave me a hint which was sufficient for ...
-2
votes
0answers
16 views

Designing a maximum energy signal

A transmission channel is constrained by allowing signals that have magnitudes ㅣSiㅣ ≤ 2Volts (a) Design a valid signal sequences si for 0≤i≤4 that has the maximum Es. (b) Compute the signal energy ...
1
vote
1answer
71 views

What is the convergence radius of $\sum_{n=0}^\infty a_nx^n$ when $\{a_{n}\}$ is s.t. $a_1 = 1, a_{n+1} = \sin(a_{n})$?

My task is this: Given a sequence $\{a_n\}$ with $a_1 = 1, a_{n+1} = \sin(a_{n}).$ (i) Show that the sequence converge and find the limit as $n\to\infty$. (ii) Show that $\sum_{n=0}^\infty a_nx^n$ ...
0
votes
3answers
140 views

Why does this formula for $e$ work? [closed]

How would one prove that the following expression approaches $e$ when $n$ approaches infinity? $$(1+1/n)^n$$ edit: $e$ is the unique positive number a where the derivative of $a^x$ is $a^x$
0
votes
3answers
23 views

Checking if a term known, exists on an infinite sequence.

Given integer $b$, how to check that $b$ exists in an infinite arithmetic sequence $S_n$, where the difference between two consecutive numbers is $d$ and $S_0 = a$? That is, there exists a positive ...
1
vote
1answer
334 views

Prove the derivative vanishes given a sequence

Suppose f is strictly increasing and continuous everywhere. Suppose further that $a_n$ is a increasing sequence and $b_n$ is a decreasing sequence both tending to $x$ such that ...
0
votes
1answer
55 views

Find this integral

If $\phi(x)$ is an arbitrary normalized function, $\mu \in \Re$, Prove that $$ \lambda \int_{-\infty}^{+\infty} dx \, \left|\sqrt{|\mu|}\cdot \phi(\mu x) \right|^2 x^n= \frac B {|\mu|^n} $$ and ...
1
vote
2answers
50 views

Prove or give a counter example $ \sum_{n=1}^{\infty } a_{2n} \ and \sum_{n=1}^{\infty } a_{2n-1} $ converge than $ \sum_{n=1}^{\infty } a_{n} $ [duplicate]

claim: if$$ \sum_{n=1}^{\infty } a_{2n} \ and \sum_{n=1}^{\infty } a_{2n-1} $$ both converge than $$ \sum_{n=1}^{\infty } a_{n} $$ converge. I managed to prove that this claim is true if $a_n ...
-1
votes
2answers
36 views

Can decreasing sequence of sets with $A_i$ containing infinitely less elements than $A_{i-1}$ have finite limit?

An updated question to one I just asked. Can we have a decreasing sequence of sets $A_n$ each a subset of the natural numbers with all members containing countably infinitely many elements such that ...
1
vote
1answer
68 views

How does one find if the following series converges: $\sum_{n=1}^{\infty} \left(1-\cos\dfrac{\pi}{n}\right)$

$$\sum_{n=1}^{\infty} \left(1-\cos\frac{π}{n}\right)$$ Its limit is $0$ so the necessary condition is verified. Now I don't know how to check whether it converges or not
2
votes
1answer
85 views

Calculating the Cesaro sum of $1-1+0+1-1+0+\dots$

I am having difficulty understanding how to find the Cesaro sum of the series: $1-1+0+1-1+0+\dots$ I know the sequence of partial sums will be: $1,0,0,1,0,0,1,0,0,1,0,0,\dots$ And hence the ...
4
votes
1answer
51 views

How to prove this continued fraction connection between $\gamma$ and $e$?

There is apparently a curious connection between Euler-Mascheroni constant $\gamma$ and $e$ in the form of an infinite series and continued fraction: $$e \gamma=e \sum_{n=1}^{\infty} ...
1
vote
0answers
36 views

Find the sum of a Cos series.

I'm using the following equation to generate a number series. $v =-c\ \cdot \ \cos \left(\frac{t}{d}\cdot \left(\frac{\pi }{2}\right)\right)+c\ +\ b$ Values I'm using to solve the series are: $b = ...
-1
votes
1answer
16 views

Buying forecast

I have a set of items. In 2013, I bought x of a certain item, in 2014 I bought y, and in 2015 I bought z. In many cases, I only bought the item in one year, like: ...
5
votes
1answer
51 views

Value of a trigonometric series [duplicate]

Question: If $x = \sin 1^\circ$, find the value of the expression: $$\frac{1}{\cos0^\circ \cos1^\circ} + \frac{1}{\cos1^\circ\cos2^\circ} + ... + \frac{1}{\cos44^\circ\cos45^\circ}$$ in terms ...
-2
votes
2answers
44 views

Proof that both series $\sum a_n$ and $\sum b_n$ are either convergent or divergent [closed]

Proof, that if $c_1a_n< b_n < c_2a_n$ with $c_1>0$ , $c_2>0$ , sequence $a_n>0$ and $b_n>0$ with for all $n>n_0$ and $n_0$ is a natural number, that both series $\sum a_n$ and ...
2
votes
0answers
39 views

Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
2
votes
2answers
19 views

Sequence of sets with limit contains finitely many elements

Can we have a decreasing sequence of sets with all members containing countably infinitely many elements whose limit only has finitely many elements? If so, what are some examples? (Not very sure ...
-2
votes
0answers
29 views

Linear algebra , magical square? [closed]

Let $T:\mathbb{R} ^{3}\rightarrow \mathbb{R} ^{3}$be a linear transformation . Prove the equivalance following statements : i) $\mathbb{R} ^{3}=ker\left ( T\right) \oplus im\left ( T\right)$ ii) ...
2
votes
1answer
42 views

Finding the value of an infinite product

Find the value of the product : $$P=\sqrt{\frac12}\sqrt{\frac12+\frac12\sqrt{\frac12}}\sqrt{\frac12+\frac12\sqrt{\frac12+\frac12\sqrt{\frac12}}}\ldots$$ This was asked in an exam yesterday, I ...
2
votes
1answer
37 views

Does this matrix series have an answer?

I'm trying to solve this series: $$\displaystyle\sum_{i=0}^{k}A^i B C^{k-i}$$ Where A, B, and C are $N\times N$ symmetric matrices. And $A$ and $C$ have spectral radii smaller than or equal to 1, ...
0
votes
2answers
25 views

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$.

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$. I have never done an example of convergence of sequences that have characteristic (indicator) ...
1
vote
0answers
25 views

Convergence in distribution for two random variables

If $\lim\limits_{n \to \infty} P(X_n\leq T)=P(X\leq T)$ and $\lim\limits_{n \to \infty} P(Y_n\leq T)=P(Y\leq T)$, where $X_1, X_2,\cdots$ and $Y_1, Y_2,\cdots$ are two sequences of random ...
0
votes
2answers
29 views

Proving a sequence defined by a recurrence relation converges

How can I prove that this iterative sequence converges to $2$? Can I use the convergence definition? $$a_{n+1} = \frac{a_n}{2} + \frac{2}{a_n},\qquad a_0 = 4 $$ Thanks for the help.
6
votes
6answers
203 views

Show that $1-\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}-\dfrac{1}{7} -\dfrac{1}{8}-\dfrac{1}{9}-\dfrac{1}{10} … $ converge

$$1-\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}-\dfrac{1}{7} -\dfrac{1}{8}-\dfrac{1}{9}-\dfrac{1}{10} ... $$ I added parentheses for each sub-sequence with the same sing. so i ...
1
vote
3answers
193 views

Proof of this (geometric) infinite sum

Question: Prove that when $\exp(\Re[st])>1$: $$\sum_{n=0}^{\infty}\exp\left[-nts\right]=\frac{e^{st}}{e^{st}-1}$$ I think that it is an geomeric sum, but I new to this kind of finding a ...
0
votes
1answer
32 views

Verify $P\left(e^{-\frac{\pi}{x}\cdot2^{2^{-k}\cdot{n}}}\right)=\left[(2x)^{2^k}\cdot{2^{-n}}\right]^{2^{2-k}}$?

Let $$P(q)=1+240\sum_{n=1}^{\infty}\frac{n^3q^n}{1-q^n}$$ We have a closed form for $q=e^{-\frac{\pi}{x}\cdot2^{2^{-k}\cdot{n}}}$ ...
0
votes
1answer
18 views

Criteria for convergent sequence (Baby Rudin Theorem 3.22)

Theorem 3.22 of Rudin's Principles of Mathematical Analysis says: $\sum a_{n}$ converges iff for every $\epsilon>0$ there is an integer $N$ such that $$|\sum_{k=n}^m a_{k}|\leq \epsilon$$ if ...
1
vote
0answers
29 views

Can this sum over the q-Pochhammer symbol be simplified?

While considering the problem of the expected value of a dice fixing strategy on a two-sided die that comes up as $1$ with a probability of $\alpha$ and $0$ otherwise. I was studying the strategy ...
0
votes
0answers
42 views

limit of complicated sum that wolfram alpha cannot solve

Let $1 \leq d\leq \frac{1}{2}\log^3(n) \sqrt{n}$. We would like to show that for any such $d$ we have $$\sum_{x=0}^{\lceil \log^6(n) d \rceil}\left(\log^3(n) \sqrt{n}\right)^{d+x} ...
-1
votes
0answers
49 views

Chaotic Sequence $X_{n+1}=4X_n(1-X_n)$

I want to examine the chaotic sequence $$\begin{cases}X_{n+1}=4X_n(1-X_n),\\ X_0=0.2\end{cases}$$ More specifically, I want to prove that: The sequence does not converge It is non periodic It does ...
0
votes
0answers
73 views

Ramanujan's Series for 1/pi - Finding 17,526,100 digits of a partial sum that coincide with 1/pi

It is kind of hard to summarize in the title alone. These questions relate to Ramanujan's series for $\frac{1}{\pi}$: $$\frac{1}{\pi} = \sum_{n=0}^{\infty} ...
2
votes
2answers
61 views

Proving the convergence/divergence of $\sum_{n=1}^\infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+…+\frac{1}{\sqrt{n}})$ [closed]

Do the following series converges? Why? $$ \sum_{n=1}^ \infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}})$$
1
vote
1answer
65 views

Is $\sum_{k=1}^\infty \dfrac{1}{k^2}+\dfrac{1}{(k+1)^2}=\sum_{k=1}^\infty \dfrac{1}{k^2}$?

I think yes, since the first would be a kind of subsequence of the partial sums of $\dfrac{1}{k^2}$... To provide some context, the question arised while studying Fourier Series on Apostol, when was ...
0
votes
0answers
14 views

Hadamard's theorem; redefining indexing variable

I have seen in a few proofs the use of Hadamard's theorem to prove convergence of series like this: $\sum_{n\geq 0}z^{n!}$, or $\sum_{n\geq 0}z^{n^2}$ through simply changing the variable of indexing ...
0
votes
4answers
29 views

Convergence when the comparison test cannot be applied

I had a standard problem in my textbook which was to determine the convergence of $\sum _{n=2}^\infty\frac{n^3+1}{n^4-1}$. To determine whether the series is convergent or not the standard solution ...
2
votes
0answers
19 views

Bijective mapping between face polytopes of permutohedra and partitions of integers

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
0
votes
1answer
44 views

Interchanging limit and double series

I have a generating function $$U(z,w)=\sum_{j=0}^{\infty}\sum_{n=0}^{\infty}u_{j,n}z^jw^n$$ Where $0<z<1$, $0<w<1$, $0\leq u_{j,n}<1$. Is is true for this series that $$\lim_{z \to ...
0
votes
3answers
28 views

How could we show that $s_n=2e^{(-n)}$, if $n$ is even & $-3\over n$ when $n$ is odd converges to $0$ as $n \to \infty$?

How could we show that $s_n=2e^{(-n)}$, if $n$ is even & $-3\over n$ when $n$ is odd converges to $0$ as $n \to \infty$ ? Using only the definition of convergence. So what I have tried is finding ...
0
votes
0answers
8 views

Trace of the heat operator $Z(t)=\sum_{m,n=1}^{\infty}e^{-\frac{\alpha_{m,n}^2}{r_0^2}t}$

I know that the spectrum of the disk of radius $r_0$ is $\lambda_{m,n}=\frac{\alpha_{m,n}^2}{r_0^2}$, where $\alpha_{m,n}$ is the n-th root of the Bessel's function of order $m$. I have to find the ...
0
votes
0answers
23 views

Pole Of Zeta function extension

Please advise on how to proceed with this? I don't know where to begin? Thanks. Show that the Riemann-Zeta function has a pole of order 1 at 1 after it has been extended to a holomorphic function on ...
3
votes
2answers
128 views

Closed form of $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{\lfloor|\tan x|\rfloor}{|\tan x|}dx$?

I am trying to find a closed form for the integral $$I=\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{\lfloor|\tan x|\rfloor}{|\tan x|}dx$$ So far, my reasoning is thus: write, by symmetry through ...
0
votes
1answer
32 views

Bound for series

Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the ...
0
votes
4answers
163 views

Summation Problem (lower limit is variable) [closed]

$$\sum_{j=i}^n 2$$ I am having difficulty solving this summation. Can i have hint or solution to this problem?
-2
votes
2answers
52 views

Limit of $a_n$ vs limit of $(-1)^na_n$ [closed]

Is it true that $\lim_{n\to\infty}a_n=0$ for some sequence $a_n>0$ if and only if $\lim_{n\to\infty}(-1)^na_n=0$ ? This seems like intuitively it would be the case; but I am unsure.