For questions about recurrence relations, convergence tests, and identifying sequences

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2answers
86 views

What is the formula to generate this number sequence : 1 , 7 , 14, 30

What is the formula to generate this number sequence : 1 , 7 , 14, 30 I'm sure this is very simple for you guys. But it's got me alittle stuck. Thanks To clarify, I'm not an advanced maths student. ...
0
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0answers
11 views

Prove the combination rule for non-negative series by using comparison test

The combination rule for non-negative series is as follows: The question is, how can I prove combination rule for non-negative series by using the comparison test, namely:
0
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1answer
22 views

summation of series by telescoping series method (feedback needed)

i am stuck i did the first part by cancelling out terms since its a telescoping series. But I do not know how I can proceed any further . Please help. I am not sure of whatever i have done so far. so ...
2
votes
3answers
113 views

Find if this series converges and if so find its value

I need help I cant understand how we can solve this. I am confused when the log came in. I listed the first few terms but i do not know how to proceed further. all I know is that the sequence is ...
0
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0answers
35 views

Study the series $\sum_{n=1}^{\infty} \left|\frac{1}{n^{\cos\theta+ i \sin \theta}}\right|$

Study the character of the series $$\sum_{n=1}^{\infty} \left|\frac{1}{n^{\cos\theta+ i \sin \theta}}\right|$$ With i is the immaginary unit, $\theta$ is a real angle. My answer is that the series ...
0
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2answers
30 views

Theorem 3.29 in Baby Rudin

Theorem 3.29 in Walter Rudin's Principles of Mathematical Analysis, 3rd ed., states that If $p>1$, then the series $$\sum_{n=2}^\infty \frac{1}{n (\log n)^p} $$ converges; if $p \leq 1$, the ...
2
votes
1answer
34 views

Definition of the limit of a sequence

I'm looking over the following definition of convergent limits: A sequence $(x_n)$ in $\mathbb{R}$ is said to converge to $x \in \mathbb{R}$, or x is said to be a limit of $(x_n)$, if for every ...
1
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2answers
62 views

What branch of analysis deals most with sequences and series?

I'm really interested in sequences and series (mainly series). What kind of math branch should I look more into? I understand that sequences and series mostly point toward analysis, but what ...
1
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1answer
20 views

Recognising that $\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$

So I know from Mathematica that: $$\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$$ I am wondering how someone could ...
0
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2answers
31 views

Prove that sequences $\frac{a_n}{b_n} = 0$

Let $(a_n)$ and $(b_n)$ be positive real sequences such that $\lim \limits_{n \to \infty} \dfrac{a_n}{b_n} = 0$ and $(b_n)$ is bounded. Prove that $\lim \limits_{n \to \infty} a_n=0$. Proof: ...
0
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1answer
27 views

Radius and Interval of Convergence of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{2n-1}$

This is my first time finding the radius and interval of convergence of a series, so please bear with me. I would like to find the radius and interval of convergence of ...
2
votes
1answer
27 views

Series and comparison test

If $a_n>0$ and $\sum a_n$ diverges, what can be said about $\displaystyle \sum \frac{a_n}{1+na_n}$? I cannot prove that it is convergent or divergent. I think it is convergent for some examples ...
0
votes
1answer
33 views

Test whether $\sum_{n=1}^{\infty}\frac{\ln{n}}{n}$ converges or diverges

I am trying to solve this using an integral test, but I am unsure whether or not this is correct. Let $f:[2,\infty)\to\mathbb{R}$ be defined by $f(t)=\frac{\ln{(t)}}{t} >0\ \forall t\geq2$. Now ...
-4
votes
2answers
51 views

Limit of a headache-giving series [on hold]

I found this problem somewhere which says to find out the limit of this series, in order to prove that the limit is somewhere outside $\mathbb{Q}$. $$ x_n = \sum_{k=0}^n 2^{-k^2-k}\;,\quad \forall n ...
1
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0answers
20 views

Recursive sequence with a formula for a part of its criteria

I have the next recursive sequence which firts terms are $$2,\ \frac{3}{2},\ \frac{10}{7},\ \frac{17}{12},\ \frac{112}{89}$$ I need to express it has a general form for the $n$th element, I can't make ...
0
votes
2answers
17 views

Limits of sequences and series

I'm currently learning about the limits of a sequence based on the following definition: A sequence $x_n$ is said to converge to $x \in \mathbb{R}$, if for every $\epsilon > 0 $ there exists a $K ...
1
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0answers
23 views

Closed form of $\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n$

Inspired by answers to this question, for which $n$ values could we specify a closed-form of $$S(n)=\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n\,?$$ Here $\psi_1$ is the trigamma function, and ...
0
votes
1answer
24 views

Radius of convergence: $\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$

It is asked to find the radius of convergence of the series $$\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$$ i.e, to find the values of x such that this series converges. Clearly, I could directly apply ...
1
vote
2answers
48 views

Why $\cos(n^2x) \not\to 0$ for any real $x$?

I'd like to show, as simply as possible, that $\sum_{n=1}^\infty\cos(n^2 x)$ diverges for every real $x$. (I know how to prove it for rational $x$. For irrational $x$, I don't know if $(n^2 x)$ is ...
0
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2answers
20 views

Determine the interval of convergence of series $\sum_{n=1}^{\infty } \frac{1}{\cos ^2(n \cdot x)+\sqrt{n}}$

So, hey. I was sincerely trying to find it by myself with Weierstrass M-test, but failed occasionally, because I ended up with $\sum_{n=1}^{\infty } \frac{1}{\sqrt{n}}$,which is a divergent series. ...
1
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0answers
14 views

Alternating series involving sums of k-primes

As an exercise, if $p_k$ are positive integers composed of k primes including repetition and $\pi_k(n)$ the number of $p_k$ not exceeding n can we show that for the alternating series of sums of ...
1
vote
1answer
35 views

Convergence of $\sum_{n=1}^{\infty} \frac{1}{n^z}$

Let us consider $z\in \mathbb C$; what is the condition on modulus of z in order that $$\sum_{n=1}^{\infty} \frac{1}{n^z}$$ the series (zeta function?) converges? For example, if $|z|=1$, the series ...
1
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2answers
45 views

how to solve the following without a calculator

Is their a method for the required question except brute force ? (here $11$ terms gets multiplied as followed) ...
0
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2answers
21 views

Is there a mistake in this page on asymptotic expansions?

I think there is an error in section 4.3 of this page - http://aofa.cs.princeton.edu/40asymptotic/ The author says that by taking $x = -\frac{1}{N}$ in the geometric series $\frac{1}{1-x} = 1 + x + ...
0
votes
3answers
36 views

How can I determine if the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges or diverges?

Determine whether the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges. If it converges find the limit and if it diverges determine whether it has an infinite limit. Proof: let $a_{n} = ...
0
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0answers
20 views

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]. Now calculate the expansion about 0. I'm having trouble doing this. I'm not even sure which is the best way to approach it, for ...
0
votes
2answers
64 views

About the rationality of $1.1010010001\dots$ [duplicate]

Let's define $\rho=1.1010010001\dots$ which can be expressed by: ...
1
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1answer
28 views

Proof that $ y(n) = ∑_{k=-∞}^{∞}\ {a}^{-k}u(n-k)u(-k) = \frac{1}{1-a }$ if $n>0$

Can someone explain the steps and how the boundaries for the summation change to result in the answer (And possible for the case where $n\leq 0$. I am not really a mathematician, don't know if the ...
2
votes
1answer
36 views

Proving the convergence of a recursive sequence

Consider the following sequence, defined recursively: $$ x_{n+1}=\frac{2x_n^3+2}{3x_n^2} $$ Prove that $x_n$ converges to $ 2^{1/3} $ and $ x_7 $ approximates $ \root 3 \of2 $ accurately to 6 ...
1
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0answers
30 views

Is a sequence always defined by a formula holding for all values?

Suppose we have the sum of a series $S_n = T_1 + T_2 + \cdots + T_n, S_1 = 6, S_2 = 20, S_n = 6S_{n-1} - 8S_{n-2}$. The explicit formula for the sum can then be derived as $S_n = 4^n + 2^n$. Yet when ...
0
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1answer
21 views

Studying the convergence of a series with logarithm [on hold]

I would like to know if this sum is convergent, and why: $$\sum_1^\infty\ln\bigg(1+\dfrac1{n^a}\bigg)$$
0
votes
5answers
53 views

Convergence/Divergence of the series $\sum\limits_{n=1}^{\infty}\tan(1/n)$

Trying to see if $$\sum\limits_{n=1}^{\infty}\tan(1/n)$$ converges or diverges. As $n \to \infty$, $\tan(1/n) \to 0$, so inconclusive. Ratio test was inconclusive, root test doesn't look good for this ...
2
votes
1answer
27 views

Other Useful Series Tests

So after taking calculus II, or maybe a first course in analysis, everyone learns a few series tests. They learn 1) Divergence Tests 2) Integral Test (from which we deduce things like $p$-series. ...
1
vote
1answer
50 views

Which is true for a sequence $(x_n)$ such that $x_n\in (o,\frac{1}{n})$?

If $(x_n)$ is a sequence of real numbers such that for every $n,x_n\in (0,\frac{1}{n})$ then which of the following is true? $1.\lim_{n\to\infty}x_n=0$ $2.$If $f$ is continuous function from ...
0
votes
2answers
30 views

Convergence of the series $\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$

I would like to see whether or not $$\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$$ is a convergent series. Root test and ratio test are both inconclusive. I tried the alternating ...
0
votes
3answers
51 views

Showing the divergence of the series where $a_1 = 2$ and $a_{n+1} = \frac{5n+1}{4n+3}a_n$.

Consider a series such that its $i$th term $a_i$ is defined by $a_1 = 2$ and $a_{n+1} = \dfrac{5n+1}{4n+3}a_n$. I would like to show that this series is divergent. Here's how I thought about it: ...
1
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0answers
29 views

Prove that the relationship exists

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
1
vote
1answer
28 views

bounding $\sum_{i=1}^m 2^{2^i \cdot k}$

I'm trying to put a nicer upper bound for $\sum_{i=1}^m 2^{2^i \cdot k}$ than $2\cdot 2^{2^m\cdot k}$ which is twice the last term of the series. (In fact it seems that $1.5 \cdot 2^{2^m \cdot k}$ ...
2
votes
0answers
27 views

sum of series with square power

Is there an explicit expression to calculate the sum of a series with a square power? \begin{equation} \sum_{k= 1}^{N}{r ^{k^2}} \ or \sum_{k= 1}^{\infty}{r ^{k^2}} \ \end{equation} regards arthur
1
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1answer
36 views

What type of series is this: $k^n + k^{n-1} + k^{n-2} + k^{n-3}+\dots$

I am wondering what type of series this this, where you have some constant (let's say 4) to the power of n, summed up where each new exponent keeps going $n-1, n-2, n-3, n-4, ...$ and so on. So, ...
1
vote
2answers
29 views

Rudin's Chapter 3: Numerical sequences and series

In the Rudin's Principles of Mathematical Analysis 3rd edition, Chapter 3, page 56, there is a definition of a set $E$ that, in my point of view, is very doubtful. What is the set $E$? I couldn't also ...
4
votes
2answers
118 views

The Series $-\frac{2}{5}+\frac{4}{6}-\frac{6}{7}+\frac{8}{8}-\frac{10}{9}-\cdots$

Stewart claims that this series is convergent, but Wolfram and I disagree. I looked at $$\lim\limits_{k\to\infty}\dfrac{(-1)^k (2k)}{4+k} $$ which is clearly not 0. Did I do something wrong?
0
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0answers
13 views

Prove the uniform convergence of series in a compact space

I have to prove that if $\sum\limits_{n=1}^\infty \frac{1}{|a_n|}$ converges, then $\sum\limits_{n=1}^\infty \frac{1}{x-a_n}$ converges absolutely and uniformly in any compact space that $\forall n ...
0
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0answers
13 views

Multiplication of Limits when both diverges

I am working on the multiplication of limits, and I am able to prove when both converge, the multiplication converges to the multiplication of limits, however I cannot grasp the idea when for instance ...
0
votes
2answers
28 views

Let $a_n\in A$ and $\lim_{n \to \infty}|a_n-a_{n+1}|=0$,then which is true?

Let $a_n\in A$ and $\lim_{n \to \infty}|a_n-a_{n+1}|=0$,then which is true? $1.$There exists $a\in A$ such that $a=\lim_{n \to \infty}a_n$ $2.$There exists $b\in \mathbb{R}$ such that $b=\lim_{n \to ...
3
votes
1answer
48 views

Convergence of $\sum x_k \log (x_k^{-1}) $ [on hold]

Prove or disprove: If $\{x_k\} \subset (0,1)$, and $\sum x_k < \infty$, then $\sum x_k \log (x_k^{-1}) < \infty$. What if $\{x_k\}$ is monotone? Thanks a lot.
0
votes
0answers
15 views

How I can explain this contradiction regarding the position of initial condition $x₀$? [on hold]

The motivation to this question can be found in http://mathworld.wolfram.com/LogisticMapR=4.html. My question: In formula (4) the initial condition $x₀$ must be in the interval $(0,1)$ while in ...
0
votes
0answers
17 views

Strong law of large numbers when sample size is a random variable

For a sequence $X_1, X_2, \ldots, X_n$ of i.i.d. random variables with mean $\mu$, the strong law of large numbers tells us that $$\sum_{i=1}^{n} \frac {X_i} {n} \xrightarrow{a.s.}\ \mu ...
1
vote
0answers
30 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
2
votes
0answers
32 views

Prove that $\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$

Put $g_n = 1 + \frac{1}{2} + ... + \frac{1}{n} - \log(n)$. Prove that $$\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$$ I've tried this for a while now ...