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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
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1answer
15 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
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3answers
55 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
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1answer
23 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 ...
0
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0answers
51 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.

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
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3answers
68 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
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1answer
29 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 $\...
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0answers
29 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
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1answer
108 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
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1answer
47 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
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1answer
31 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
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1answer
16 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
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4answers
68 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
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2answers
72 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 ...
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3answers
40 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
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2answers
50 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
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4answers
53 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
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0answers
41 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
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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 ...
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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{(...
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0answers
31 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: ...
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2answers
21 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
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2answers
55 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_{...
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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
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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 ...
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0answers
38 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
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2answers
84 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
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3answers
42 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 ...
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0answers
33 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
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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
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1answer
39 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
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2answers
45 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 ...
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1answer
22 views

series convergence imply nx approaches 0

Proof: For a decreasing sequence of positive reals, show that if the sum converges, then $nx_n \to 0$ but the converse is not true The first part I just assumed a positive limit the series converge ...
3
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0answers
63 views

Is there any closed expression for $\sum_{k=0}^\infty r^{k^2}$?

Let $r$ be any real number with $0 < r < 1$. Then, of course, there is a closed expression for $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$ But does there also exist a closed expression for $$\sum_{...
1
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2answers
34 views

Proof that $\bigcap_{n\in\mathbb{N}}[a_n,b_n]$ is a non-emtpy set

Let $a_n,b_n \in \mathbb{R}$, for $n\in \mathbb N$ with $a_n \leq a_{n+1} \leq b_{n+1} \leq b_n$. Proof that $\bigcap_{n\in\mathbb{N}}[a_n,b_n]$ is a non-emtpy set. My attempt: Observe $A:=\{a_n : ...
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1answer
24 views

Show a function defined on rationals integrable

Let q1,q2,... be a fixed enumeration of rationals in [0,1]. Define f(x) so $f(q_n)=a_n $ for all a and 0 otherwise. Prove that it is Riemann integrable if the sequence an approaches 0, and justify ...
2
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2answers
51 views

Find $\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}$ [duplicate]

How could one find $$\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}\,?$$ I tried to use Fourier series and integrals depending on a parameter to reduce the problem to a differential equation, but that ...
1
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0answers
76 views

If $\sum a_n$ converges, then $\sum a_n^{\frac{n}{n+1}}$ converges as well.

If $(a_n)$ be a sequence of positive real numbers such that $\sum a_n$ converges, then show that $\sum a_n^{\frac{n}{n+1}}$ converges as well. Attempt: We try to apply the Limit form of the ...
2
votes
3answers
64 views

Evaluate the limit $ \lim_\limits{n \to \infty} \frac{1^a+2^a\cdots+n^a}{n^{a+1}} $ [duplicate]

The exact question is- Find the real value(s) of $a (a \ne -1)$ for which the limit $$ \lim_{ n \to \infty} \frac{ 1^a+2^a\cdots+n^a}{(n+1)^{a-1}[ (na+1)+(na+2) \cdots+(na+n)]} = \frac{1}{60}$$ I ...
1
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2answers
37 views

Converges or diverges $\sum_{n=1}^\infty \frac{\ln n}{\sqrt n}$?

I was trying to find if the series $\sum_{n=1}^\infty \frac{\ln n}{\sqrt n}$converges or diverges. First, I tried ratio test and got the limit as 1. I tried Limit Comparison Test's and I only got 0's ...
-1
votes
1answer
22 views

Limit of a recursive sequence defined by the average of previous terms. [on hold]

Let $x_0=0,~x_1=1$ and define $x_{n+1}=\frac{x_{n-1}+x_{n}}{2}$ for all $n\geq 1$, then how can we prove $x_n$ converges to $\frac{2}{3}$
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3answers
48 views

Sin(x+h) Taylor’s series. Is ‘h’ in degrees or radian? [on hold]

In the $\sin(x+h)$ Taylor’s series, can $h$ be in degrees or it has to be in radians?
3
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2answers
61 views

$\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in…

Let $\{a_n\}$ be a sequence of positive real numbers such that $a_1 =1,\ \ a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$. Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in......
1
vote
0answers
30 views

Calculate the series.

There is given series $\sum_{n=1}^{\infty}\frac{1}{(n(n+r))^{a}}$ Where $r$ is natural number, and $a$ is a reral number big enough so that this series is convergent. Calculate the value of this ...
1
vote
0answers
19 views

Convergence acceleration of a series by using transformation

One of the ways of accelerating the convergence of a series is by transforming into a faster series. Examples of this approach can be found in this paper. A generalization of this method leads to the ...
0
votes
3answers
58 views

Creating a formula for $a_n$ for a Fibonacci like sequence

This is sequence is "Fibonacci like": $$t_1, t_2, t_1+t_2, t_1+2t_2,...$$ How can I find the $1001^{st}$ term of this sequence. I'm a littler confused because this sequence is neither arithmetic ...
0
votes
2answers
43 views

Infinite Sum of Series

So I was given this question $$T_n = \sum _ {k=0}^{ n-1} \frac{n}{n^2+kn+ k^2} $$ And $$S_n = \sum _{ k=1}^n \frac{n}{n^2+kn+ k^2} $$ We were asked wether $T_n$ or$S_n$is$ \gt$or$ \lt \frac{π}{3\...
0
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0answers
30 views

Finite series with sine in the denominator

The given question: $$\displaystyle \sum_{k=1}^{13} \dfrac{1}{\sin \left(\frac{\pi}{4} + \frac{(k-1)\pi}{6}\right) \sin \left(\frac{\pi}{4} + \frac{k\pi}{6}\right) }=?$$ By expanding the ...
-1
votes
2answers
32 views

Determine the series is conditionally or absolutely convergent. [on hold]

This is the problem: $$\sum_{n=1}^\infty (-\frac{1}{2})^n$$ How can we decide the series is a conditionally or an absolutely convergent?
1
vote
2answers
41 views

For what value of r and p is the series convergent $\sum _{ n=1 }^{ \infty } \frac{r^n}{n^p}$

I have been given the series $\sum _{ n=1 }^{ \infty } \frac{r^n}{n^p}$, where $r, p > 0 $ Which seems to be a combination of a geometric series and a p-series. The summation of geometric series ...
5
votes
2answers
55 views

Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function. $\int_{a}^{\infty} f$ converges.Prove that $\lim_{x\to\infty} f(x)=0$

Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function in that range. $\int_{a}^{\infty} f$ converges. Prove that $\lim_{x\to\infty} f(x)=0$ Hint: Use the sequence $F_n(x)=n\...