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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.

0
votes
2answers
20 views

How to find the limit sum of a series [on hold]

Suppose $$S_n = \lim_{n\to\infty}\frac{\exp(i/2)}{\sum_{j=1}^{i}\exp(j/2)} \ \ \ \text{where} \ \ i = 1,\ldots,n$$ Programatically, $S_n\approx 0.49$ but would I show this by hand?
-4
votes
1answer
41 views

What is the next number in this series? 1, 2 ,3, 22, 4, 23, 5, 222, … [on hold]

I just know there has to be a pattern and that it has something to do with primes.
0
votes
1answer
30 views

Prove by induction that, for all $n\in\Bbb N$, $\sqrt{n} ≤ \sum_ {k=1}^n \frac{1}{\sqrt{k}} < \sqrt{n} + \frac{n}{\sqrt{n+1}}$.

So, I know that for my base case I use $n=1$, and that for the inductive hypothesis we assume the pattern holds until the $n-th$ iteration. Then use that to prove the $(n+1)-th$ iteration ($\Bbb P(n)\...
0
votes
1answer
14 views

Recognizing a Factoring Pattern (Pt. 2)

I am trying to identify a pattern in the following set of equations; $N_{-1}=1$ $N_{0}=2y$ $N_{1}=2y^2+z$ $N_{2}=2y^3+3yz$ $N_{3}=2y^4+5y^2 z+z^2$ $N_{4}=2y^5+7y^3 z+4yz^2$ $N_{5}=2y^6+9y^4 z+...
4
votes
6answers
57 views

Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$

Let $$x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_{n}})$$ Prove that $x_{n+1} < x_{n}$ for $a \geq 0$. Hint: Let the initial guess satisfy $x_{1} > \sqrt{a}$ I am stuck at how to begin this. I ...
-1
votes
0answers
33 views

How to express this function as power series? [on hold]

How to express this function as power series $\frac{x}{(2+x^2)^2}$
-4
votes
2answers
31 views

Determine the series whether convergence or divergence with using ratio rest. [on hold]

This is the problem: $$\sum_{n=0}^\infty 3^n\sin((\frac{1}{4})^n)$$ I can't prove the convergence of this series, how can we solve it?
0
votes
0answers
52 views

turning $2x$ into a perfect even

So I am trying to generate a sequence with an equation (that I don't think exists) and it involves all the even numbers, and one way to find the sequence is to get rid of all odd prime numbers so... $...
1
vote
2answers
47 views

Evaluation of series $\sum_{n=0}^\infty\frac{5n+1}{(2n+1)!}$

How to evaluate series $$\sum_{n=0}^\infty\frac{5n+1}{(2n+1)!}$$ I tried to split the summation...but I failed. Please help
1
vote
0answers
45 views

Question about big $O$ notation

We all know that exponential functions grow faster than polynomials. Let us consider the following function: $f(n) = n^{a_1} \cdot (\log n)^{a_2}\cdot (\log \log n)^{a_3} \cdot (\log \log \log n)^{a_4}...
1
vote
1answer
20 views

Couple of questions on Hurwitz theorem

Hurwitz theorem as stated in Hahn and Epstein's Classical Complex Analysis is as follows: I've got couple of questions I'm stuck on. Is the converse of the Hurwitz theorem is true ? Provide an ...
0
votes
3answers
37 views

How to prove the double sum of combinations is $3^n$

I have a double sum of combinations as follow $$S = \sum_{i=0}^{n}\sum_{k=i}^{n}{n \choose k}{k \choose i}.$$ I guessed and tested that $S = 3^n$, but I have no idea how to prove this. Any help is ...
0
votes
1answer
13 views

Is the proportionality characteristic of this function being carried on?

$A=kx$ is a directly proportional function,where $A^2=B^2+C^2$.Does it necessarily mean $B$ and $C$ both vary directly with respect to $x$? If not, under what condition is this possible? Thank you in ...
0
votes
0answers
8 views

Legendre polynomial expansion of a positive function

Let´s assume I have a function $f(\theta)>0$ defined for $\theta<\pi$ and $\theta >0$. I want to find its Legendre polynomial decomposition $f(\theta)= \sum_{l=0}^\infty f_l \, P_l(\cos{(\...
3
votes
1answer
41 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
0
votes
1answer
46 views

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ [duplicate]

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if (1). $0<a<e$ (2). $0<a\leq e$ (3). $0<a<\frac{1}{e}$ (4). $0<a\leq \frac{1}{e}$ I tried ...
0
votes
5answers
34 views

Sequence divergence test

Could I argue that $\left(1+\frac{1}{n}\right)^{n^2}$ = $e ^n$ therefore the sequence diverges? I am wondering if it is a legal move
2
votes
2answers
36 views

Why does $\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}$ converge?

I am looking through examples on convergences of random series, and in one of the proofs the following result is used: If $\epsilon > 0$ then $$\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}&...
0
votes
1answer
42 views

Recognising a Factoring Pattern

I am trying to identify a pattern in the following set of equations; $$N_0=y$$ $$N_1=y^2+z$$ $$N_2=y^3+2yz$$ $$N_3=y^4+3y^2z+z^2$$ $$N_4=y^5+4y^3z+3yz^2$$ $$N_5=y^6+5y^4z+6y^2 z^2+z^3$$ Essentially, ...
0
votes
1answer
17 views

Is there a special name for numbers whose multiples remain multiples when reversed?

I recently noticed that reversing multiples of 11 gives you other multiples of 11, for example 11 x 19 = 209 & 902 ÷ 11 = 82 ...is there a name for that? &, if so, does that same name ...
1
vote
3answers
74 views

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!} $ converge?

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!} $ converge? I have no idea how to do this. I have tried to use any trick I am aware of but can't figure this out. Can anyone help ...
0
votes
1answer
33 views

Struggling with finding a potential counterexample for a convergent series.

This question comes with two parts. Part (a): Let $\{f_n(x)\}$ be a sequence of nonnegative functions for $x \in S \subseteq \mathbb{R}$ such that $f_1 \geq f_2 \geq \dots \geq 0$, and that $f_n \to ...
-1
votes
0answers
64 views

Let $f_n$ is continuous function from $[0,1]$ to $\mathbb{R}$ for any natural $n$. And for any $x$ from $[0, 1]$ series $\sum_n f_n(x)$ converge. [on hold]

Prove that exist a positive length interval $[a, b]$ in $[0, 1]$ such that partial sums of series $f_n$ is evently limited in $[a, b].$
3
votes
3answers
77 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
0
votes
1answer
30 views

Convergence of a sequence… [duplicate]

Let $\{a_n\}$ be a sequence of real numbers. Define $\sigma_n = 1/n(a_1 + \dots + a_n)$. Suppose that $\lim a_n = a \in \mathbb{R}$. Show that $\lim \sigma_n = a$. Here is my work so far... Fix $\...
-1
votes
0answers
48 views

Help calculate the limits

Help calculate the limits: 1) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{\sqrt {k(n-k)}}$$ 2) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{(n-k)\ln{n}} $$ 3) $$\lim_{n\to \infty}{n}^p \sin(\pi(\sqrt 2 ...
3
votes
1answer
109 views

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. Show $\lim_{n \rightarrow \infty} x_{n}$ exists. [duplicate]

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. show $\lim_{n \rightarrow \infty} x_{n}$ exists. To do this the problem has been broken down into three pieces: a) Show that $x_{n} <...
2
votes
1answer
50 views

Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Here is Theorem 30.2 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a topological space. (a) Let $A$ be a subset of $X$. If there is a sequence of points of $A$ converging ...
1
vote
1answer
32 views

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence?

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence? My attempt via induction: If I prove that the denominator grows faster than the numerator, I can conclude ...
0
votes
1answer
19 views

the taylor series of the function $f(x) = A/(x-B)^4$ using geometric series.

I have to find the taylor series of the function $f(x) = A/(x-B)^4$ using geometric series. If rewrote it to the general geometric series $\sum x^n=\frac{1}{1-x}$ $A\frac{1}{x-B}=A\frac{-\frac{1}{B}}...
2
votes
4answers
69 views

Does the series $\sum\limits_{n=0}^{\infty}e^{-n}$ converge? [on hold]

I came upon this question , $\sum\limits_{n=0}^{\infty}e^{-n}$ converge ? where as no description of n is given , I tried to use something called power series , but here n is negative can I ...
0
votes
2answers
73 views

what is $\lim_{x\to 0} f(x)$

Let $$f(x)=\sum_{n=1}^{\infty}{\sin(nx)\over n^2}$$ Then what is $\lim_{x\to 0} f(x)$ Now I know the series converges uniformly by $M-Test$ (Take $M_n=1/n^2$). What should be my next step. I am ...
-1
votes
3answers
44 views

How to solve this expression

$$\sum_{n=1}^\infty\frac1{(x-3)^{2n-1}}$$ How to change the given expression to a rational function
-4
votes
2answers
54 views

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ [on hold]

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ We have $-1\leq\cos{x}\leq 1$. So $(\cos{x})^n \to 0$ as $n\to \infty$ Please solve this problem. Please find the sum.
2
votes
4answers
55 views

Limit of the converging infinite sequence $\frac{2e^{3n}-1}{e^{3n}+1}$.

What is the limit of $$\left\{\dfrac{2e^{3n}-1}{e^{3n}+1}\right\}_{n=1}^{\infty}$$ How do I deal with the 'n' when it is the power of e? What steps would I take and is there any site / video that ...
1
vote
0answers
46 views

Converting an Arithmetic Series to Sigma Notation

I've been struggling with the following problem for quite a while now, and have been unable to identify a pattern; You have a geometric series $Y$ for which we have the following rule: $$Y_{t+1} ...
0
votes
1answer
39 views

Finding the general term of a number pattern

This is the number pattern 15, 29, 56, 108, 208, ... The pattern is as follows, Term 1 = $15$ , Term 2 = Term $1 \times 2 - 1 = 15 \times 2 - 1 = 29 $, Term 3 = Term $2 \times 2 - 2 = 29 \times ...
0
votes
0answers
43 views

Proof-Verification: Find $\lim\limits_{n \to \infty}x_n$ where $x_n=\frac{n^k}{a^n}$

Problem Let $x_n=\dfrac{n^k}{a^n}$($a,k$ are constants and $a>1$). Find $\lim\limits_{n \to \infty}x_n$. Solution Notice that $$\lim_{n \to \infty}\frac{x_{n+1}}{x_n}=\lim_{n \to \infty}\frac{(...
0
votes
0answers
32 views

If $\sum a_n$ is convergent but not absolutely, then $\sum a_n^+$ diverges

Let $a_n \in \mathbb{R}$, such that $\sum_{n=1}^\infty|a_n|= \infty$ and $\sum_{n=1}^m a_n \to a$, as $m \to \infty$. Let $a_n^+=\max\{a_n,0\}.$ Show that $\sum_{n=1}^\infty a_n^+= \infty$. Approach: ...
0
votes
2answers
23 views

Uniform convergence of series of function

The series of functions $f_n(x)= x^n/(1+x^n)$ is uniformly convergent on $[0,a]$ where $0<a<1$ and not uniformly on $[0,1)$. I have come across lots of problems like this , where the open ...
1
vote
2answers
63 views

convergence of a fibonacci-like sequence

I posted a question earlier on finding a formula for the sequence $$t_1, t_2, t_1+t_2, t_1+2t_2,....$$ This is the question I posted earlier I want to show that as $n\rightarrow \infty$, $\frac{t_{...
0
votes
5answers
30 views

what's the best way to prove the equivalences of such formulas?

I want to prove the following: $$2^n+2^{n-1}+...+2^1 + 1 = 2^{n+1}-1$$ The only Method that I know of is proof-by-induction but is this the best way to prove the equivalences of such formulas?
1
vote
1answer
20 views

Optimally order circular sequence to minimize sum of consecutive differences

We have given array $A$ of size $n$. We should find order of array $A$ that minimizes the following sum $\sum_{1\leq i \leq n}|A_i - A_{i+1}|$. We assume that $A_{n+1} = A_1$, just to simplify the ...
0
votes
0answers
40 views

A limit tends to Euler's constant

Is this limit obvious that it is tended to Euler's constant? $$\lim_{n \to \infty}\sqrt{\sum_{k=1}^{n}H_{k}\left(\frac{1}{k}+\frac{1}{k+1}\right)}-\ln n=\gamma$$ Where $H_k$ is the Harmonic number ...
7
votes
2answers
88 views

A riddle involving series.

Father has left to his children several identical gold coins. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and ...
1
vote
3answers
43 views

Convergence of $\sum_{n=2}^\infty \frac{(\ln n)^3}{n^2}$

I was trying to find if the series $\sum_{n=2}^\infty \frac{(\ln n)^3}{n^2}$ converges or diverges. But I couldn't solve the question and I looked at the solution in here. In that page, Limit ...
0
votes
0answers
35 views

How to find the general term of this recursive sequence? [duplicate]

$$ (a_n)_{a \ge 1}, a_{n+1} = (n+1)a_n + 1, a_1 = 1 $$ I found the first 5 terms and I got 1, 3, 10, 41, 206, but I don't know what to do next? I tries OEIS but it just shows me the recursive ...
0
votes
2answers
43 views

Does $\sum_{i=2}^n \frac{3}{n\ln(n)}$ converge or diverge? [on hold]

I came upon this question while working: $$\sum_{n=2}^\infty \frac{3}{n\ln(n)}$$ And I was wondering whether it converges or diverges? A help would be greatly appreciated ! Thank you!
1
vote
1answer
40 views

Determining the Values of $\alpha$ for Which the Series is Conditionally and Absolutely Convergent

The task is to determine for which values of $\alpha$ is the following series is conditionally convergent and absolutely convergent. My attempt is below. $$\sum_{n=1}^{\infty} {n^{-\alpha}\cdot(\ln{n}...
1
vote
2answers
46 views

$X_i$, $i=1,2,..n$ independent R.Vs $P(X_i=1)=\frac{3}{4} ,\ P(X_i=-1)=\frac{1}{4}$. Prove $\sum_{i=0}^nX_i \to \infty$ a. s. as $n \to \infty$

I am asked to prove $X_1+X_2+X_3+...+X_n$ diverges almost surely as $n \to \infty$ Let $Y_n=X_1+X_2+...+X_n$ then what we want to prove is $P(Y_n=k)=1, \text{ as} (k,n) \to (\infty,\infty)$ Let us ...