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

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5
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1answer
100 views

Series of Functions and Continuity

Let $a > 0$, and $(f_n)_{n=0}^{\infty}$ a sequence of continuous functions $f_n:[-a,a] \rightarrow \mathbb{R}$. Assume that the series \begin{equation} \sum_{n=0}^{\infty} x^n f_n(t) \end{equation} ...
1
vote
2answers
45 views

How do you work out the product of this sequence?

I got a question in my maths paper and I didn't know how to answer it. This was the question: What is: $(1+\frac{1}{2}) (1+\frac{1}{3}) (1+\frac{1}{4}) $... All the way up to 98 factors a) What is ...
0
votes
3answers
65 views

How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$

How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$? Can anybody shed light on this? regards.
0
votes
0answers
21 views

Where were my mistakes when I've combined $\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$ and the Fourier series for the fractional part?

Let $s=x+it$ the complex variable, thus we are denoting $\Re s=x$. Combining the identity $$\eta(s)=\left(1-2^{1-s}\right)\zeta(s),$$ that holds for $0<x<1$, where $\zeta(s)$ is the Riemann Zeta ...
2
votes
2answers
51 views

nature of the series $\sum (-1)^{n}n^{-\tan\left(\tfrac{\pi}{4}+\tfrac{1}{n} \right)}$

I would like to study the nature of the following serie: $$\sum_{n\geq 0}\ (-1)^{n}n^{-\tan\left(\dfrac{\pi}{4}+\dfrac{1}{n} \right)} $$ we can use simply this question : Show : $(-1)^{n}n^{-\tan\...
2
votes
2answers
48 views

Help with Telescopic Series with 3 terms in denominator

All the examples i have done and seen only have 2 terms in the denominator so I am a bit stuck with this one. I have attached what I have done so far, not sure how to proceed with it. Thank you for ...
4
votes
0answers
225 views

Explicit solutions to a digamma function equation

My main question: Can we obtain the exact solutions from the following equation? $$ \sum_{k=1}^{n}\cfrac{1}{k-x-1}=0 $$ Notation: This problem was reached from the digamma function $\psi$ as ...
1
vote
1answer
16 views

Bounds for coefficients of bounded infinite series

Assume that $\left|\sum_{n=0}^\infty a_n\frac{\lambda^n}{n!}e^{-\lambda}\right|\leq1$ holds for all $\lambda\geq 0$, do the coefficients $a_n\in\mathbb{R}$ need to be bounded, e.g. $|a_n|\leq 1$?
0
votes
0answers
42 views

what is the asymptotic expansion of this $_2F_2$ function?

We need to expand the function $_2F_2(a+b x,1; 1+a+b x, b x; x)$ near $x=+\infty$. Where $a$ is complex, $b>1$. When $x\to+\infty$, both the parameters and the variable goes to infinity, we can ...
3
votes
1answer
91 views

Is my proof ok? If $\sum u_n$ diverges then $\sum \frac {u_n} {u_1 + u_2 + \dots + u_n}$ also diverges

The question is : If $\sum u_n$ is a divergent series of positive real numbers and $s_n = u_1 + u_2 + \dots + u_n$ , prove that the series $\sum \frac {u_n} {s_n}$ is divergent. I tried my best. ...
1
vote
3answers
78 views

Does the sequence $T_n=2^{-n}$ converge?

$$T_n=2^{-n}$$ How can I tell if this converges? with previous questions I have just let $n = \infty$, however I'm unsure about this one.
0
votes
1answer
57 views

How prove that $\frac{a_{4n}-1}{a_{2n+1}}$ is integer

I would appreciate if somebody could help me with the following problem. Q: How prove that $\frac{a_{4n}-1}{a_{2n+1}}$ is integer ? $$a_{n+2}=a_{n+1}+a_{n}, a_1=1, a_2=1$$ I tried to solve by ...
1
vote
1answer
35 views

Find the value of $\sum_{i=1}^n \left(\frac{14i}{n}-5\right)\frac{4}{n}$ as an expression involving $n$

$\sum_{i=1}^n \left(\frac{14i}{n}-5\right)\frac{4}{n}$ and where $\sum_{i=1}^n i=1+2+3+\cdots+n=\frac{n(n+1)}{2}$ Not entirely sure if I did this correctly, but I basically plugged in $\frac{n(n+1)}{...
4
votes
3answers
153 views
5
votes
3answers
2k views

Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$)

Most of us are aware of the famous "Basel Problem": $$\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$ I remember reading an elegant proof for this using complex numbers to help find the value of ...
1
vote
3answers
90 views

How to prove $\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right)$?

I need help, on how to prove $$\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right).$$ Any hints?
6
votes
1answer
245 views

Proving a necessary and sufficient condition for compactness of a subset of $\ell^p$

Let $A \subset \ell^p$, where $1 \le p \lt \infty$. Suppose the following conditions are true: 1) $A$ is closed and bounded 2) $\forall \epsilon \gt 0, \: \exists \: N \in \mathbb{N}$ such ...
425
votes
34answers
49k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
2
votes
2answers
52 views

Proof related to the least squares method

I've seen this exercise in several statistics text, but how they get to the final formula is something that I don't quite get. How do two squared terms suddenly become a binomial term? I've been ...
0
votes
0answers
33 views

Method for writing set intersections/unions/complements/etc. in terms of polynomial functions? [closed]

I realize that I'm woefully uninformed regarding most math concepts, and I also realize (given the aforementioned fact) that I'm probably not using the correct format/notation, so please, be gentle. ...
0
votes
0answers
58 views

What is the sum of $\sum_{i=0}^\infty \frac{i^n}{4^i}$ in terms of n? [closed]

$n$ is an non-negative integer. If it is not computable, explain why? Question 1.6d of Data structures and algorithm analysis by mark allen wiess. I am able to calculate this sum for n=0,1,2,3.. but I ...
0
votes
4answers
118 views

Determine if $\sum_{n=1}^{\infty} \frac{5^{n+1}}{(3n-2)!}$ is convergent.

Determine if the following series is convergent: $$\sum_{n=1}^{\infty} \frac{5^{n+1}}{(3n-2)!}$$ so we have $$u_n=\frac{5^{n+1}}{(3n-2)!}$$ $$u_{n+1}=\frac{5^{n+2}}{(3n+1)!}.$$ With the ratio test: ...
1
vote
1answer
119 views

Taylor expansion of $f(x) = \frac{x}{x+3}\frac{1}{x-2}$ near $x=2$.

I am trying to Taylor expand the function $$f(x) = \frac{x}{x+3}\frac{1}{x-2}$$ around the point $x_0 = 2$. Clearly, the last factor explodes around this point, so I will try and expand that term. ...
0
votes
0answers
33 views

which denominator will keep the ratio?

I'm given three numbers :$a_1>a_2>a_3$ now I'm told theyre divided by natural numbers $n_1,n_2,n_3$ such that $1>a_1/n_1>a_2/n_2>a_3/n_3>0$ . (the $n$'s could.be any natural numbers ...
-1
votes
1answer
21 views

Geometric progression and logarithms

I would like to ask you for some help, solving that: 'The sum of three members of a geometric progression ($a, aq, aq^2$) is $62$ and the sum of their decimal logarithms $lg$ is equal to $3$. $a$ and ...
1
vote
1answer
27 views

Can you provide us an asymptotic for this series involving Mertens functions?

Let for integers $k\geq 1$, the Möbius function denoted by $\mu(k)$, and $M(n)=\sum_{k\leq n}\mu(k)$ the Mertens function, then one can prove easily that $$\sum_{k=1}^n\mu(k)\frac{e^{\mu(k)}+1}{e^{\...
3
votes
2answers
44 views

Given $\frac{x_n}{ x_{n+1}} \leq \frac{y_n}{ y_{n+1}}$ and $\sum y_n$ converges, prove $\sum x_n$ converges

With $x_n>0, y_n>0$, $\exists N $ such that $$\forall n>N:\quad \frac{x_n}{ x_{n+1}} \leq \frac{y_n}{ y_{n+1}}$$ Want to prove: If $\sum y_n$ converges, $\sum x_n$ converges. I think this ...
7
votes
3answers
6k views

Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$

Let $\{s_n\}$ and $\{t_n\}$ be sequences. I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo: $\limsup\limits_{n \...
8
votes
4answers
370 views

Does the series $1-\frac12+\frac12-\frac1{2^2}+\frac13-\frac1{2^3}+\frac14-\frac1{2^4}+\frac15-\frac1{2^5}+\cdots$ converge or diverge?

$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{2^2}+\frac{1}{3}-\frac{1}{2^3}+\frac{1}{4}-\frac{1}{2^4}+\frac{1}{5}-\frac{1}{2^5}+\cdots$ I've be trying to figure out how to write this series symbolically so I ...
5
votes
3answers
252 views

Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?
1
vote
0answers
56 views

Let $u_n > 0 ,v_n > 0$ for all $n$ which are bounded, then $(\lim \sup u_n)\cdot(\lim \sup v_n) \geq \lim \sup u_n v_n$.

The question is : Let $\{u_n\}$ and $\{v_n\}$ be two bounded sequences such that $u_n > 0$ and $v_n > 0$ for all $n \in \mathbb {N}$, then show that $(lim \sup u_n).(lim \sup v_n) \geq lim \...
3
votes
1answer
49 views

Power Series in Two Variables and Radius of Convergence

Let $\alpha > 0$, $\beta > 0$, and assume that the power series with real coefficients \begin{equation} \sum_{n,m = 0}^{\infty} a_{n,m} x^{n} y^{m} \end{equation} is absolutely convergent for ...
1
vote
1answer
60 views

Does $\sum \frac{1}{n^x}$ converge uniformly?

$\sum \frac{1}{n^x}$ for $x \in \left[\frac{4}{\pi},\infty\right)$ converge uniformly. I'm trying to use the Weierstrass M-Test but having trouble finding an $M_n$. Any hints?
1
vote
1answer
45 views

Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $

As the question says, $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$ where a is a constant, $a>0$.
0
votes
3answers
66 views

Confused computing sum of Fourier series

I am having some issues understanding Fourier series and I am stuck trying to solve a problem. So the function $u$ has period $2\pi$ and is defined as $$u(x) = \begin{cases} 1 & 0 \leq x \...
4
votes
1answer
86 views

$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$. Then find $\lim_{n \to \infty} a_n$.

Let $\{a_n\}$ be a sequence of real numbers such that $$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$$ Then find $\lim_{n \to \infty} a_n$. Tried very hard yet not ...
2
votes
2answers
52 views

Cantor's theorem about countable sets

Let $f:\mathbb{N} \rightarrow \mathbb{R}$ be a sequence of real numbers. Cantor's theorem states that no interval (a,b) can be the range of a sequence of real numbers. I know the proof for this ...
4
votes
3answers
168 views

Convergence of $\sum ( \cos \sqrt[3]{n^3 + \sqrt n + 7} - \cos \sqrt[3]{n^3 - 2\sqrt n + 3})$

I have some problem with this example: $$\displaystyle \sum_{n=2}^{\infty}\Bigg(\cos\Big(\sqrt[3]{n^3+\sqrt{n}+7}\Big) -\cos\Big(\sqrt[3]{n^3-2\sqrt{n}+3}\Big)\Bigg)$$ the only idea that crossed my ...
12
votes
1answer
335 views

A Variation on the Coin Problem

Suppose I have a sequence $a_n$, whose entries are the ordered elements of $S_{x,y}$: $S_{x, y}= \{ z \mid \left( z=n_1x+n_2y \right) \wedge \left( n_1, n_2 \in \mathbb{N}_1 \right) \wedge \left( \...
12
votes
1answer
259 views

Can different choices of regulator assign different values to the same divergent series?

Physicists often assign a finite value to a divergent series $\sum_{n=0}^\infty a_n$ via the following regularization scheme: they find a sequence of analytic functions $f_n(z)$ such that $f_n(0) = ...
2
votes
3answers
64 views

Why does changing the center of a geometric power series change the interval of convergence?

I know that the interval of convergence of the geometric power series $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ is $(-1,1)$. Why is it that if I do the following manipulation $$\frac{1}{1-x}=\frac{1}{...
0
votes
2answers
80 views

About bound of series

$$ Can anyone please tell me bound for following series I simplified this upto $$(2n-1)-2\left( \frac{1}{n+1}+\frac{2}{n+2}+\ldots+\frac{n-1}{2n-1} \right)$$ Also I get one of my bound as $\frac{n(...
2
votes
0answers
33 views

Asymptotic expansion of elliptic integral

I am trying to find the first 2-3 terms of the asymptotic expansion in terms of 1/ρ of the elliptic integral \begin{equation} I_n(\rho)=\int_0^\frac{h_2}{\rho}\frac{t^{2n}/h_2^{2n}}{(E_n(t))^2\...
1
vote
0answers
31 views

Question About Cauchy Product;

Is there a formula to multiply many series (More than two) using the Cauchy product? If there isn't, please tell me how I can write this formula $ \left( \frac{1}{a-e^x} \right) ^{n+1}$as the ...
0
votes
2answers
60 views

Determine convergence or divergence of the series

I have the following series : $$\sum_{k=1}^{\infty }\left ( \frac{k}{\sqrt{4k^{3}+1}} \right )$$ And I am trying to see if it's convergent or divergent. I first thought about the integral test, but ...
1
vote
0answers
41 views

Show that $\sum (-1)^n x^{(2^n)}$ has no limit as $x \uparrow 1$

Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$ I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the ...
1
vote
2answers
43 views

Prove $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ is uniformly convergent [duplicate]

I'm completely confused by uniform convergence, but I put together the following proof just based on my other questions here and examples I read online. Discussion: Let $\epsilon \gt 0$ We want to ...
2
votes
0answers
18 views

An interesting connection between the Möbius function and the parity of the number of sublattices of index $n$ in generic $3$-dimensional lattice

I recently discovered an interesting connection between the following two On-Line Encyclopedia of Integer Sequences (OEIS) sequences: A001001 and A209635. More specifically, there seems to be an ...
1
vote
0answers
64 views

A series involving digamma function

I am trying to solve the series $$\sum_{k=1}^\infty\frac{1}{k(k^2+n^2)}$$ The best I got is $$\frac{\Re\left\{\psi(1+in) \right\}+\gamma)}{n^2}$$ I am not able to simplify it more. Maybe there ...
195
votes
17answers
14k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...