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Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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3answers
49 views

If $x_1=\sqrt 2$ and $x_{n+1}=(\sqrt2)^{x_n}$ then sequence $x_n$ converges to 2.

If $x_1=\sqrt 2$ and $x_{n+1}=(\sqrt2)^{x_n}$ then show that sequence $x_n$ converges to 2. I know this sequence is monotonically increasing. But how to prove it converges to 2? The sequence is ...
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4answers
40 views

Prove that $\sum_{n=1}^∞\frac{\left(\ln n\right)^3}{n^3}$ is a convergent series by using comparison test

I proved by using the integral test that the series is convergent but can't find a way to prove by using the comparison test, which was required.
1
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1answer
26 views

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove the following properties.

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove: 1) If $\alpha > 0$, show $\sum_{n = 0}^{\infty} a_{n}x^{n}$ converges if $|x| < 1/\alpha$ and diverges if $|x| > 1/\alpha$ ...
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1answer
25 views

Formula for analytic functions?

In here (third under double infinite series) they list the following formula. $$\displaystyle \sum_{k=0}^{\infty} \sum_{j=0}^{\infty} a_{k,j} = \sum_{j=0}^{\infty} \sum_{k=0}^{j} a_{k, j-k}$$ Is ...
0
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0answers
26 views

Convergence of the series $\sum_{n=1}^{\infty} a^{\ln(n)}$ [duplicate]

Let us consider the series $$\sum_{n=1}^{\infty} a^{ln(n)}$$ Now my question is: for which value of $a$ the series will converge, provided that $a>0$. I have tried with $\textbf{D Alembert's ...
0
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1answer
26 views

How can I show $f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}$ is continuously differentiable on $x \in ]-1,+1[$ for each $n \in \mathbb{N}$?

I want to show $f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}$ is continuously differentiable for each $x \in ]-1,+1[$ and $n \in \mathbb{N}$? I have $$f'_{n}(x) = \frac{x}{\sqrt{x^{2} + \frac{1}{n}}}.$$ ...
1
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1answer
33 views

For each interval $[a,b]$ contained in $I$, sequence $\{f_{n}:[a,b]\rightarrow\mathbb{R}\}$ converges uniformly to $f : [a,b] \rightarrow\mathbb{R}$

$I$ is an open interval Using the following fact to show this: ${\{f_n\}}$ converges pointwise on $I$ to the function $f$, and ${\{f'_n}\}$ converges uniformly on $I$ to the function $g$ Attempt: ...
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0answers
39 views

A question from G.Polya

A question from G.Polya: prove $$1 + \frac 1 4 + \frac 1 9 + \frac 1 {16} + \cdots = \log x \cdot \log (1 - x) + \frac {x+(1-x)} {1} + \frac {x^2 + (1-x)^2}{4}+\frac{x^3+(1-x)^3}{9}+\cdots$$ ...
0
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1answer
33 views

Proving $\sum_{k = 0}^{\infty} \frac{1}{1 + |x|^{k}}$ converges if and only if $|x| > 1$

I would like to show $$\sum_{k = 0}^{\infty}\frac{1}{1 + |x|^{k}} $$ converges if and only if $|x| > 1$. I think that the best way to show the backwards direction is to assume we havve $|x| \leq ...
2
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1answer
23 views

Proving the sequence $f_{n} = \sqrt{x^{2} + 1/n}$ converges uniformly to $f(x) = |x|$ on $(-1, 1)$.

I have the following exercise from my book: For each $n \in \mathbb{N}$ and each $x\in (-1, 1),$ define $$f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}$$ and define $f(x) = |x|$. Prove that the ...
2
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3answers
44 views

When is a sum of continous functions continous?

Let $(f_{\alpha})_{\alpha \in A}$ be a family of real valued continous functions on a topological space $X$ such that for each $x\in X$ $f_\alpha (x)\neq 0$ for only finitely many $\alpha\in A$. Then ...
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1answer
48 views

Questions on two Formulas for $\zeta(s)$

This question is related to the following two formulas for $\zeta(s)$. (1) $\quad\zeta(s)=\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^\infty\frac{1}{2^{n+1}}\sum\limits_{k=0}^n\frac{(-1)^k\binom{n}{k}}{(k+1)...
2
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0answers
29 views

Convergence of a series involving cosine

Let $x \in (0, 2\pi)$. Is the series $\sum_{n=1}^{\infty} \frac{\cos(n^2x)}{n}$ convergent? My guess is: YES and I would like to use Dirichlet test: however I have troubles proving that the partial ...
1
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1answer
22 views

Formulated a Series Problem But Unfortunately Don't Know How To Solve

This all started when I was playing around with a financial spreadsheet. There is no need to know financial terms as I've managed to convert this observation into a mathematical problem. But just in ...
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2answers
33 views

How this series a_j converges?

We have $a_n\geq 0$ and suppose that $\sum_{j=n}^{2n} a_j\leq \frac{1}{\sqrt{n}}$. I dont know how to derive that $\sum a_j$ converges.
1
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1answer
20 views

Prove the complex series converge absolutely

Prove the complex $$\sum_{k=0}^\infty (k+k^2i)^{-1}$$ series converge absolutely Solution: by triangle inequality, we have $$|k+k^2i|\ge k^2-k \ge {k^2\over 2}$$ if $$k \ge 2 $$ how to understand ...
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1answer
24 views

Taylor polynomial Approximation of a value using degree 3

How would I use a Taylor polynomial of degree 3 to approximate 33^(1/5). Can you please go in detail with your steps.
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2answers
33 views

Why do I have to use L'Hopital in limit comparison test for $\sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right)$

I'm trying to determine whether the following series diverges or converges: $$\sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right)$$ I use the limit comparison test, comparing it to $\frac{1}{n}$, which ...
1
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4answers
39 views

Proving that the power series for the cosine function is greater than zero, for $x$ in $[0, \pi/2)$.

I'm trying to prove the cosine power series $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} \;>\;0$$ for all $x \in [0, \pi/2)$. Here, $\pi$ is defined as the smallest positive real such that $...
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2answers
17 views

A question on the sums of finite subsequences of a sequence of positive reals

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers such that $a_n\rightarrow 0$ and $\sum\limits_{i=1}^{\infty}a_i$ is divergent. Prove that the set containing the sums of all $finite$ ...
0
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3answers
45 views

Series $\sum(-1)^{a(n)}\frac1{a(n)}$ converging to infinity with $a:\mathbb N\to\mathbb N$ a bijection

EDIT: I am terribly sorry everyone, I wrote geometric series instead of harmonic. Sorry for that. I also briefly scanned the answers and I am not sure anymore the result would be a harmonic series ...
1
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0answers
39 views

Convergence of $b_n = \frac{\sum_{i=1}^na_i}{4}$

Question: Let $\{a_n\}_{n\in \mathbb N}$ be a sequence of real numbers, and for each $n\in \mathbb N$ define $$b_n= \frac{\sum_{i=1}^na_i}{4}.$$ Prove that if $\{a_n\}$ converges to $A$, then so ...
0
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1answer
19 views

Given $\lim_{n\to\infty}x_n = \infty$ show that $y_n = \left\{ \sum_{k=1}^n x_k\right\}$ is an unbounded sequence.

Given: $$ \lim_{n\to\infty}x_n = \infty $$ show that $$ y_n = \left\{ \sum_{k=1}^n x_k\right\} $$ is an unbounded sequence. Intuitively this is obvious, however I'm having a hard time ...
2
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0answers
35 views

Prove $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\ln{(2n+1)}}{2n+1}=\pi/4(\gamma-\ln{\pi})+\pi\ln{(\Gamma(3/4))}$

In the title, $\gamma$ is the Euler-Mascheroni constant and $\Gamma(3/4)$ represents the extension of the factorial function. This isn't a homework question or something, someone left it on a board ...
0
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2answers
35 views

Showing that $\lim_{n \rightarrow \infty} \sum_{k=1}^{\infty}k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2}))=\infty$

Does anyone know how to show $ \lim_{n\rightarrow\infty} \sum_{k=1}^{\infty}k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2}))=\infty$? My thoughts were that you should you a series approximation of some sort, but ...
2
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2answers
53 views

Closed form expression for $\frac 1{2^n} \sum_{i=0}^n \binom{n}{i}(2i-n)^{2k}$

Let $$F (n,k)=\frac {1}{2^n}\sum_{i=0}^n \binom{n}{i}(2i-n)^{2k},$$ where $n,k$ are non-negative integers. By numerical tests the expression is an integer polynomial in $n $ of order $k $: $$ F(n,0)=...
1
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2answers
37 views

Average power of 2 in all even natural numbers

Consider all even natural numbers. Every 4th number has a power of 4 (or $2^2$) Every 8th number has a power of 8 (or $2^3$) Every 16th number has a power of 16 (or $2^4$) What is the average ...
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2answers
45 views

Closed form for $\sum_{n=1}^\infty \log(n) * x^n$

As in the title, I'm in quest for $\sum_{n=1}^\infty \log(n)\cdot x^n$, where $0 \le x \lt 1$ Wolfram Alpha says: $-\operatorname{PolyLog}^{(1, 0)}(0, x)$, but I don't understand what that means. (...
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1answer
22 views

Identity theorem for a holomorphic funtion defined near zero

I have to show, whether there is a holomorphic funtion $f$ defined in an open neighborhood of zero, such that: $$ f\left(\frac{1}{n}\right)=(-1)^n \frac{1}{n^3}$$ for all positive integer $n$. My ...
1
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1answer
20 views

Prove that F is a bounded linear functional and $||F|| _{X^*}=||w||_{\infty}$.

Let $(X,||\cdot||)=(l^1,||\cdot||_1)$, $w=(w_n)_{n \geq 1} \in l^{\infty}$. Define for all $x=(a_n)_{n \geq 1}\in l^{1}$, $$F(x)=\sum_{n \geq 1}w_na_n.$$ Prove that $F$ is a bounded linear functional ...
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2answers
88 views

Find the $\lim_{n\to\infty}n^{2} \int_{0}^{1} \frac{1}{(1+x^{2})^{n}} \, dx$. [on hold]

Find the $$\lim_{n\to\infty}n^{2} \int_{0}^{1} \frac{1}{(1+x^{2})^{n}} \, dx.$$I don't know how to find integral of $ \int_{0}^{1} \frac{1}{(1+x^{2})^{n}} \, dx$. Please help!
2
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1answer
25 views

A sequence of thirteen integers…if every selection of twelve terms from the sequence contains six terms…

This is a delightful Oxford entrance question from 1993, but I'm stuck on the final bit. A sequence $$N_1, N_2, N_3, ..., N_{13}$$ of thirteen integers is said to be lucky if every selection of ...
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2answers
25 views

Mathematical Statistics : Transformation in a sequence of real random variable

How to solve this problem : Let $(X_k)_{k \in N^*}$ be a sequence of i.i.d. real random variables distributed as a Bernoulli of parameter $p \in (0,1)$. For each $k \in N^*$, we set $Y_k = X_k X_{k+1}...
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0answers
17 views

Is the (x-a) format necessary when finding the radius of convergence for a geometric series?

So I am taking AP Calculus BC, and we are currently working on convergence and divergence of series. I came across the following problem in one of my homework assignments: Here is the work I did to ...
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1answer
19 views

Given $\lim_{n\to\infty}x_n = x \ne 0$ and $\lim_{n\to\infty}y_n = \pm\infty$ show that $\lim x_ny_n = \mp\infty$ when $x<0$

Given $$ \begin{cases} \lim_{n\to\infty}x_n = x \ne 0 \\ \lim_{n\to\infty}y_n = \pm\infty \end{cases} $$ Show that for $x < 0$: $$ \lim x_ny_n = \mp\infty $$ I've started with the case when ...
8
votes
4answers
143 views

Sequence such that every subsequence can have a different real limit [duplicate]

I would like to find a sequence of real numbers $(a_n)_{n\in\mathbb{N}}$ with this property: for any $L\in\mathbb{R}$ there is a subsequence $a_{k_n}$ such that $$\lim_{n\to\infty} a_{k_n} = L$$ Does ...
-1
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6answers
44 views

If $0< a < 1$; show that $\lim na^n$ goes to $0$. [duplicate]

If $0< a < 1$; show that $\lim na^n$ goes to $0$ as $n$ goes to $\infty$ I know that when $0< a < 1$, $\lim a^n$ goes to $0$ and $\lim n$ goes to $\infty$ $|a^n| <\epsilon$ take $\...
3
votes
2answers
189 views

Determine if this specific sequence is a Cauchy sequence

I have the following sequence: $$a_n =\sum_{k = 1}^n (-1)^{b_k} {1\over k^2} $$ And the hint is that I have to prove that: $$ {1\over k^2} < {1\over k-1} - {1\over k} $$ So assuming $m>n$, I ...
1
vote
3answers
87 views

The series $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+…$

Consider the expression $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+...$ Denote the numerator and the denominator of the $j^\text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, ...
0
votes
2answers
40 views

Calculate the limit of the limits of sequences

Suposse the sequence $x^{n}$ such that $$x^{n}= (x^{n}_1, x^{n}_2,..., x^{n}_m,...),\quad 0 \leq x^{n}_m \leq 1, \forall n,m \in \mathbb{N}$$ $$\lim_{m\to \infty} x^{n}_m = 1, \quad \forall n \in \...
1
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1answer
45 views

Limit of sequences - including infinite limits

I am in need of finding limits for the following sequences: $$a_n = \bigg(1+ \frac1n \bigg)^{n^2} $$ $$b_n = (-1)^n (\sqrt[n]n)$$ $$c_n = 2^n - \bigg(2+ \frac1n \bigg)^n $$ $$d_n = \bigg(\sqrt[n]n + {...
0
votes
2answers
36 views

Questions about conditionally convergent series and rearrangement of [on hold]

According to Riemann Series Theorem or Riemann Rearrangement Theorem a conditionally convergent series - with a clever rearrangement of terms - can converge to any desired value, or even can be shown ...
1
vote
1answer
33 views

Let $A$ be $10 \times 10$ real matrix. then which of the following is correct?

Let $A$ be $10 \times 10$ real matrix. then which of the following is correct? [$\rho(A)=Rank(A)]$ (A) $\rho(A^8)=\rho(A^9)$ (B)$\rho(A^9)=\rho(A^{10})$ (C) $\rho(A^{10})=\rho(A^{11})$ (D)$\rho(A^...
0
votes
1answer
43 views

Divergence of the sequence $\{\frac{e^n}{2^n+1}\}$

While studying sequences I came across $a_n= \frac{e^n}{2^n+1}$ which is divergent according to the answer key. Attempting to take the limit results in $\lim_{n \to \infty}\frac{e^n/2^n}{1+1/2^n}$. ...
0
votes
1answer
13 views

Uniform Convergence on the middle 1/3 of an interval.

I came across this problem when studying up on uniform convergence and for the life of me I haven't been able to work it out. Especially with the odd requirement for the interval of uniform ...
1
vote
1answer
32 views

A sequence of rationals converging to an irrational point (proving that $\mathbb Q$ is not locally compact)

Here is my attempt to prove that $\mathbb Q$ is not locally compact. (My questions are below the proof.) Suppose $\mathbb Q$ is locally compact. Then it is locally compact at every point. Let $x\in \...
0
votes
1answer
33 views

Finding general formula for a recursively defined series [on hold]

Can someone please show me how to solve the following question please, I am very lost. Suppose that the sequence $x_0, x_1, x_2...$ is defined by $x_0 = 6$, $x_1 = 2$, and $x_{k+2} = −6x_{k+1}−8x_k$ ...
1
vote
3answers
46 views

Why is a divergent/convergent series multiplied with a constant a divergent/convergent series again?

I was trying really hard to find a series smaller than $\sum\limits_{k=1}^{\infty} \frac{1}{2k}$ to prove, that $\sum\limits_{k=1}^{\infty} \frac{1}{2k}$ is divergent. Now I got to know that I can ...
1
vote
2answers
40 views

Solving a difference equation for coin toss sequence probabilities

I want to solve the following difference equation: $$b_n-b_{n-1} = \frac{1}{8}(1-b_{n-3})$$ I tried to solve it similar to the solution of Fibonacci sequence here, but when I try to assume the ...
1
vote
3answers
45 views

Using $\tan^{-1}$ Show that

$$ \pi = 2 \sqrt3 \sum_{n=0}^\infty {(-1)^n\over(2n+1)3^n}$$ Really have no idea on this one guys. Its a practice question for my calc 2 final. Please help.