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

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4
votes
3answers
43 views

Series - Convergence or divergence - $a_n = \frac{(n!)^2}{2^{n^2}}$

I have this series: $$a_n = \frac{(n!)^2}{2^{n^2}}$$ I tried to solve it with: $$\lim_{n\to\infty} \frac{a_n+1}{a_n}$$ So I get: $$\lim_{n\to\infty} ...
1
vote
0answers
49 views

Weierstrass theorem and the series $\sum\frac{z^k}{k^2}$ and its derivative

The Weierstrass theorem states that if a series with analytic terms, $f(z)=f_{1}(z)+f_{2}(z)+\cdots +f_{k}(z)+\cdots $ converges uniformly on every compact subset of a region $\Omega$, then the sum ...
0
votes
0answers
29 views

Infinite sequences with all but finitely many elements equal

Consider two infinite sequences $\{a_1,a_2,\dots\}$ and $\{b_1,b_2,\dots\}$ such that $0<|a_i|<1$, $0<|b_i|<1$, $|a_i|\geq|b_i|$ for all $i$ and $a_i=b_i$ for all except finitely many $i$. ...
0
votes
1answer
49 views

Regarding absolute continuity of some function

$f (y) $ is continuous function of y. $\int_{-\infty}^\infty |f(y)||(x-y)|^2dy$ is finite for all x Given $h(x)= \int_{-\infty}^\infty f(y)(x-y)^2dy=\int_{-\infty}^\infty f(y+x)(y)^2dy$ is $h(x)$ ...
1
vote
2answers
46 views

Computing an infinite sum using the Residue theorem

Show that $$\sum_{n=-\infty}^\infty \frac{1}{(3n-1)^2} = \frac{4\pi^2}{27}$$ Here is what I tried so far. I know that I can use the Residue theorem to solve a summation of this form. ...
3
votes
0answers
27 views

Limit of infinite sum with asymptotic limits

Suppose that $a_n$ is some sequence, and let $f,g:\mathbb{N}\to\mathbb{N}$ such that $f(n) \leq g(n)$ for all $n$, and that $\lim\limits_{n\to\infty}\frac{g(n)}{f(n)} = 1$. Is it true that $$ ...
1
vote
2answers
78 views

Another of $\frac{1^2}{1^2}+\frac{1-2^2+3^2-4^2}{1+2^2+3^2+4^2}+\cdots=\frac{\pi}{4}$ type expressable in cube?

Gregory and Leibniz formula (1) $$-\sum_{m=1}^{\infty}\frac{(-1)^m}{2m-1}=\frac{\pi}{4}$$ We found another series equivalent to (1) This is expressed in term of square numbers ...
1
vote
2answers
51 views

Proving the fractional equation: $\{2^{n-1}\sqrt{3}\}=0.b_nb_{n+1}\ldots_{(2)}$

Prove that $$\{2^{n-1}\sqrt{3}\}=0.b_nb_{n+1}\ldots_{(2)}$$ where $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$. (Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.) I am not ...
3
votes
4answers
49 views

Is the sum of an infinite series of elements in the span of an orthonormal set also in that set?

If $(e_k)$ is an orthonormal sequence in some Hilbert space $H$ does it follow that, if for a set of scalars $\{\alpha_k\}$, the series $$\sum_{k=1}^{\infty}\alpha_ke_k$$ converges to an $x \in H$, ...
4
votes
6answers
827 views

Find the sum of the infinite series $\sum n(n+1)/n!$

How do find the sum of the series till infinity? $$ \frac{2}{1!}+\frac{2+4}{2!}+\frac{2+4+6}{3!}+\frac{2+4+6+8}{4!}+\cdots$$ I know that it gets reduced to $$\sum\limits_{n=1}^∞ \frac{n(n+1)}{n!}$$ ...
3
votes
2answers
52 views

Grouping terms of a series with the same sign

Let $\displaystyle \sum_{n=1}^{\infty} a_n$ be a series with infinitely many positive terms and infinitely many negative terms, and let $\displaystyle \sum_{n=1}^{\infty} b_n$ be the series obtained ...
0
votes
0answers
27 views

$\mid-1+2-3+4-5+6-7+10-11+…+(p_{k}-1)-p_{k}\mid=k$? Where $p_{k}$ is the $k$-th prime.

I'm not sure if it's a Telescoping series but I tried the generating rule to prove and test the series but I'm not getting any insight and I got stuck. Here are few Examples: ...
-9
votes
1answer
46 views

Convergence of a sequence defined by a recurrence relation. [closed]

Let $(u_n)_{n\geq 1}$ be the sequence defined by $$ u_{n+1} = \frac{6(1+u_n)}{7+u_n}, \quad n\geq 1 $$ and $u_1 = 1$. Show that $(u_n)_{n\geq 1}$ is increasing. Does $(u_n)_{n\geq 1}$ converge? If ...
6
votes
4answers
189 views

Limit of $\sqrt{\frac{\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$ when $x\to 1^-$?

I am trying to understand if $$\sqrt{\frac{2\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$$ is convergent for $x\to 1^-$. Any help? Update: Given the insightful comments below, it is ...
-1
votes
1answer
64 views

How to prove that if $\sum a_{n}^2$ converges, $\sum |a_{n}|$ also converges? [closed]

I can prove the converse, but not this one. Actually, it's a prove or disprove question. Any hints?
2
votes
2answers
60 views

A sequence is defined as $a_{n+1}=1+\frac{1}{a_n}$ for $n \ge 1$ with $a_{1}=1$

A sequence is defined as $a_{n+1}=1+\frac{1}{a_n}$ for $n \ge 1$ with $a_{1}=1$ Will it converge and to what limit? I found that the hint given in the book is that interpret each term ...
2
votes
1answer
36 views

Can you provide us a good approximation for $\sum_{n=1}^{\infty} \left| \log \left( 1+\frac{\mu(n)}{n^2} \right) \right|$?

Let $a_n=\frac{\mu(n)}{n^2}$, where $\mu(n)$ is the Möbius function. Since $\sum \left| a_n \right| $ is convergent by the comparison test, then a proposition from analysis ensures that ...
0
votes
1answer
23 views

Upper bound for $\sum_{x=1}^l \left(\frac{s}{x}\right)^x$

Let $l,s$ be some large numbers (if it helps, you might assume $s \gg l \gg 1$) and consider $$S:=\sum_{x=1}^l \left(\frac{s}{x}\right)^x.$$ What one can easily do is the following: $$S \leq ...
0
votes
2answers
34 views

Finding the number of Maxima

The given sequence $\frac{1000^n}{n!}$ , where n = 1,2,3..... a.) Does not have a maximum. b.) Attains a maximum at exactly one value of n. c.) Attains a maximum at exactly two values of n. d.) ...
3
votes
4answers
94 views

Summation of a trigonometric series - $\frac{\sin n}{n}$

Question: Find the sum of the series: $$\lim_{n \to \infty}\frac{\sin1}{1}+\frac{\sin2}{2}+\frac{\sin3}{3}+...+\frac{\sin n}{n}$$ I have no clue how to find this. Obviously I can see the sum will be ...
1
vote
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: ...
8
votes
1answer
63 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
58 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
333 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
202 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 ...