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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.

5
votes
0answers
13 views

About $\sum_{r=1}^{k}\exp\left[2i\pi\left(r+\frac{r^2}2+\frac{r^3}3+…+\frac{r^n}n\right)\right]$

Context: I recently saw user @David's profile picture and description: "My icon is the graph of the exponential sum $$\sum_{n=1}^{10620}e^{2\pi if(n)}$$ for $$f(n)=\frac{n}{20}+\frac{n^2}{9}+\frac{...
0
votes
3answers
17 views

Prove that no function $f : Z → {1, …, 100} $ is one-to-one.

This seems obvious to me, but I'm not sure how I would prove it. Is simply proving $|Z| > |{1...100}| $ sufficient? If so, how would I go about proving that? I know Cantor's theorem that says some ...
0
votes
0answers
47 views

Bolzano Weierstrass theorem

I couldn't find this anywhere on SE. The Bolzano Weierstrass theorem says that Every sequence in a closed and bounded set $S$ in $R$ n has a convergent subsequence (which converges to a point in S). ...
1
vote
1answer
29 views

Infinite solutions to a function.

I have rewritten this question. Take a series, $$ S_k = \sum_{n=1}^k u_n $$ where $S_k \in \mathbb{C}$ forms a divergent series as $k \to \infty$. Now take a function $f_n(x)$ such that there are ...
1
vote
1answer
21 views

Finding the first three nonzero terms in the Maclaurin series: $y=\frac{x}{\sin(x)}$

As the title says I would like to find the first three nonzero terms in the Maclaurin series $$y=\frac{x}{\sin(x)}$$ I have the first few terms for the expansion for $\sin(x)=x-\frac{x^3}{6}+\frac{x^...
0
votes
1answer
25 views

Infinite series question involving integrals

Can someone help me solve this tricky infinite series problem? I tried to find the indefinite integral of the nth term but my solution didn't make sense at all. I suspect there must be an ...
3
votes
2answers
24 views

Infinite divergent series - does the convergence of quotient imply convergence of difference?

I have two series with all terms positive, $$\sum_{i=1}^{n}a_i \equiv A_n >0,\;\; \sum_{i=1}^{n}b_i\equiv B_n>0.$$ Each series diverge as $n\to \infty$. We also have $A_n \leq B_n, \forall n$. ...
0
votes
0answers
22 views

Absolute value theorem for sequences

In the proof for this theorem, my textbook states: $$-|a_n|\le a_n\le |a_n|$$ for all n. And uses this as the basis for the rest of the proof. However, I cannot seem to get the intuition. For ...
2
votes
2answers
32 views

Sum and Product inversion

Under what conditions on $a_{i,j}$ when $i\in\{1,2...,n\}$ and $j\in\{1,2...,m\}$ this relation holds: $$\sum_{i=1}^{n}{\prod_{j=1}^{m}{a_{i,j}}}= \prod_{j=1}^{m}{\sum_{i=1}^{n}{a_{i,j}}}$$ Addendum ...
1
vote
0answers
31 views

Does this explicit formula for the prime-counting function $\pi(x)$ converge?

This question is related to an answer I posted earlier at the following link. Explicit Formula for $\pi(x)$ A potential explicit formula for the fundamental prime-counting function $\pi(x)$ is ...
0
votes
2answers
33 views

Find the sum of the infinite series in Calculus [on hold]

How do I find the sum of the following infinite series? $$11 + 2 + \frac{4}{11} + \frac{8}{121} + \cdots$$
0
votes
3answers
38 views

Does $\sum_{i=2}^\infty \frac{1}{(\ln(n))^2}$ converge or diverge?

I've tried, the limit comparison test with several values and have tried finding some values for the direct comparison test but nothing really concrete has come out of it. $$\sum_{i=2}^\infty \frac{...
0
votes
2answers
33 views

Solve functional equation $f(z)=c+zf(z^2)$ with series expansion?

Let the functional equation $(1)$ be given as $$ f(z)=c+zf(z^2) \tag{1}$$ where $c \in\mathbb R$ and $c \neq 0$. How can this functional equation be solved with series expansion (power, Taylor or ...
4
votes
3answers
178 views

Find general formula for the terms

$${a_n}={\frac{11}{7},\frac{107}{49},\frac{659}{343},\frac{4883}{2401},\frac{33371}{16807},\frac{234569}{117649},\dots}$$ I don't know how to approach this question as it is not arithmetic or ...
0
votes
1answer
18 views

A specific and interesting recurrence relation

Let f and g be increasing functions such that the sets {f(1),f(2),...} and {g(1),g(2),...} partition the positive integers. Suppose that f and g are related by the condition g(n)=f(f(n))+1 for all $n&...
1
vote
1answer
51 views

Prove that this sequence of continued fractions $\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}},\dots$ tends to $1$.

The Problem: I'll write up a couple more terms: $$\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}}, \frac{20}{19+\frac{18}{17+\frac{16}{15+\frac{14}{13}}}}, \frac{30}{29+\...
0
votes
1answer
49 views

How do I eliminate the other wrong answers?

Let $\{u_n\}$ be a sequence of real umbers satisfying the following conditions: (1)$(-1)^nu_n\ge 0$ (2)$|u_{n+1}|<\frac{|u_n|}{2}$, for all $n\ge 13$. Which of the following statements ...
0
votes
1answer
44 views

Let $n_1,\ldots,n_k$ be positive integers summing to $N$. What's an upper bound for $\sum_{i=1}^k1/\sqrt{n_i}$?

Disclaimer. Sorry, I haven't looked into this one in any detail (as I should have). I was just thinking there out-of-be an elementary principle out here (pigeon-hole, Cauchy-Schwarz, Jensen, etc.). ...
0
votes
0answers
29 views

Rewrite a cubic summation [on hold]

how do you write $$\left(\sum_{i=1}^{k}{ i^3}\right) + (k+1)^3$$ as a single summation?
1
vote
1answer
44 views

Give a closed formula for the recursive series of $S_1 = \frac{a_1}{b_1}, S_{n+1} = \frac{a_{n+1}}{b_{n+1}+S_n}$

The Problem: The following real numbers are given: $$a_1,a_2,a_3,...\in\Bbb{R}\backslash\{0\} \\ b_1,b_2,b_3,...\in\Bbb{R}\backslash\{0\}$$ We define a recursive series of: $$S_1 = \frac{a_1}{b_1} ...
0
votes
1answer
43 views

Can we find or construct a decreasing subsequence in $(x_{n})_{n≥1}$ converging to $0$

Let $(x_{n})_{n≥1}$ be a real sequence converging to $0$. My question: Can we find or construct a decreasing or increasing subsequence in $(x_{n})_{n≥1}$ converging to $0$.
2
votes
2answers
42 views

Generating function of binomial coefficients

We want to evaluate the sum $$\sum_{L=0}^{\infty}\frac{1}{2}L(L+1)x^L$$ From this set of notes (page 2, equation 8) we find the formula $$\sum_{n=0}^{\infty}\binom{n}{k}y^n = \frac{y^n}{(1-y)^{n+1}}$$...
1
vote
1answer
36 views

Which one is the correct notation?

I know this notation is correct: $a_1,a_2, a_3,\cdots,a_n=\left\{a_k\right\}_{k=1}^n$ Now, we have a function $f(n)$. I want to write this sequence in correct notation: $\left\{ f(1),f(2),f(3),\...
2
votes
1answer
42 views

Finding $\cos(\theta)+\cos(\theta+\alpha)+\cos(\theta+2\alpha)+…+\cos(\theta+n\alpha)$ with complex variable analysis [duplicate]

We have a series as $\cos(\theta)+\cos(\theta+\alpha)+\cos(\theta+2\alpha)+...+\cos(\theta+n\alpha)=U$ How can we make use of complex variable analysis to arrive at the term below which is ...
0
votes
0answers
42 views

The algebraic properties of a sequence

Take the sequence $S$ to be $4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13,...$. Clearly the odd indices of the sequence are elements of $4\mathbb{N}^+$, so the odd indices of $S$ form a group without ...
2
votes
2answers
52 views

Power Series expansion of $x\over(1+x-2x^2)$

I am unable to solve this specific problem. The only "notable series expansion" I can use (and know) is $\sum^{+\infty}_0 x^n =$$1\over(1-x)$ I tried several things but none worked. Writing $x\over(...
0
votes
1answer
40 views

Find $b_n$ independent of $x$ satisfying $\sum_{n=0}^{\infty}\frac{b_n}{x^{n+1}}=0$

I am looking for non-trivial real coefficients $b_n$ independent of $x$ satisfying:$$\displaystyle\sum_{n=0}^{\infty}\frac{b_n}{x^{n+1}}=0$$ or prove that $b_n=0$ for every $n=0,1,2...$ Note that $...
0
votes
1answer
19 views

Convergence of series $a(n)$, where $a(n+1) = p + qa(n)$, $n \geq 1$

Suppose for $n \geq 1$, we have for some constants $p,q$ $a(n+1) = p + qa(n)$. Conditions on $p$ and $q$ for which the series converges? So, we have the series as $a(1)+a(2)+a(3)+\ldots $ $a(1) +...
0
votes
0answers
69 views

pointwise vs uniform convergence (Baby Rudin 7.4)

This is a basic question about the relationship between pointwise and uniform convergence. Suppose $f(x) = \Sigma_{n=1} ^ \infty \frac{1}{1+n^2x}.$ The question (Rudin 7.4) is what intervals does ...
2
votes
2answers
67 views

Compute $\sum_{i,j,k=0}^\infty\frac1{3^i3^j3^k}$ with $i\ne j\ne k$

$$\sum_{i=0}^\infty\sum_{j=0}^\infty\sum_{k=0}^\infty\frac1{3^i3^j3^k}\\i\ne j\ne k$$ I could not understand how to incorporate the fact i not equal to j not equal to k
1
vote
0answers
48 views

When is $a(n)$ prime?

Question: When is $a(n)\in P$ compared to all possible values of $n$? where $P$ denotes the set of primes. What is the density of the primes in the sequence? Consider the sum of the prime counting ...
-1
votes
0answers
16 views

HW series determined where convergence or divergence where k is constant k>0

For the series {1/K^lnn Where K constant K>0 Show that this series is Converge if k>e Divergent if k<=e Let k =infinity *e
1
vote
4answers
57 views

How do we know $\sum_{i=0}^n\frac{x^n}{n!} $ converges to $ e^x $ for all x? [duplicate]

$$\sum_{i=0}^n\frac{x^n}{n!} $$ I know the sum converges for all x but how do we know it converges to the expect value $e^x$. This sum was derived as the Taylor series of $e^x$ around $0$. How do we ...
-2
votes
3answers
67 views

Need hint in evaluating $\lim_{x\to 0}(\log\frac{1}{x})^x$

What is the limit of $$\lim_{x\to 0}\Big(\log\frac{1}{x}\Big)^x$$ How can I proceed? I tried putting $x=e^{\log(x)}$ but it did not help. I would like to have some hints. Please don't provide the ...
0
votes
2answers
23 views

Finding Values for x for which $\sum_{n=1}^\infty \frac{x^n}{3^n}$ converges

My question is to find the values of x for which $\sum_{n=1}^\infty \frac{x^n}{3^n}$ converges and to also find the sum of the series for those values of x. I was going to use the ratio test, however ...
1
vote
0answers
64 views

Find the sum of the series $\sum\frac{(-1)^n}{(2n)!+1}$

The whole question looks like- Prove that, $\sum_{n\ge0}\frac{(-1)^n}{(2n)!+1}$ is convergent. Find its value. (Options: $\pi/2$, $4\pi$, $\pi/4$, $2\pi$) I have showed the convergence part. ...
0
votes
0answers
37 views

Does this sum uniformly converge? And how to calculate $\int_0^{\pi/2} \sin (\sin x) dx$?

Does this sum $$\sum_{n = 0}^{\infty} \frac{(-1)^n (\sin x)^{2n+1}}{(2n+1)!}$$ uniformly converge on $[0,2\pi]$? At the previous exercise of this exercise, I calculate $$\int_0^{\pi / 2} (\sin x)^{2n+...
-2
votes
1answer
33 views

How can I find a general term for this recursive sequence?

I'm currently studying calculus and I came across this function: the power series $\sum_{n=0}^\infty a_nx^n$, where $a_0=1$, and $a_n=\frac{6}{n}*a_{n-1}$. I'm trying to find a general term, so I ...
0
votes
2answers
22 views

find the kth term from the nth partial sum

Okay this is a very stupid question but i dont know why I dont get it so im sorry in advance the expression for the nth partial sum of a series $$\sum_{k=1}^\infty u_k$$ is $$ s_n = {(3n^2 - 1)}$$ ...
0
votes
1answer
9 views

Convergence of the Cauchy product of two series of real numbers

During these days, I reading a paper about analysis. I am confused about following question of a Cauchy-product of two convergent series. Let $\sum_{n=0}^{\infty}a_n$ and $\sum_{n=0}^{\infty}b_n$ be ...
0
votes
3answers
54 views

Showing that $\sum_{n=2}^\infty \frac{2}{n^2-1}$ is convergent [duplicate]

I am trying to show that $$\sum_{n=2}^\infty \frac{2}{n^2-1}$$ is convergent by using telescoping sum. So far I have reduced $\frac{2}{n^2-1}$ via partial fractions to $\frac{1}{n-1} - \frac{1}{n+1}$....
1
vote
1answer
38 views

A sequence defined by another sequence

Let $(a_n)_{n\ge 1}$ be a sequence of positive integers greater or equal than $2$. Consider the sequence $b_n=1-\frac{1}{a_1}+\frac{1}{a_1 a_2}-...+(-1)^n\frac{1}{a_1 a_2...a_n}$, $n=1,2,3...$. Prove ...
1
vote
2answers
61 views

Using induction to prove a sequence $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$ is increasing

I have a sequence defined by $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$. I need to use induction to show that the sequence is increasing and $a_n<3$ for all $n$. Also to deduce that $a_n$ is ...
1
vote
2answers
63 views

Series $a_n$ converges implies series of function $f(a_n)$ converges

Let $f : \mathbb{R} \to \mathbb{R} $ has the following property: For every series $\sum_{n=1}^{\infty} a_n$ converges implies $\sum_{n=1}^{\infty} f(a_n)$ converges. Show that there exist $M>0$ ...
2
votes
0answers
38 views

What are alternative ways to show this infinite sum is the absolute error between $\pi$ and 22/7

tl:dr at the end. In my first year undergraduate course having definite integrals, I studied about the $\beta$ and $\Gamma$ functions. There was a problem from this particular exam which I never had a ...
6
votes
2answers
222 views

Find Maximum of any discrete function (not necessarily a PDF)

How can we find the maximum of any discrete function, say $$ f(n)=\frac{(n+1)^2}{2^n},\quad n\in \mathbb{N} $$ that is not the PDF of any distribution? (This query is unrelated to statistics.) By ...
-1
votes
0answers
40 views

Find the sum of infinite series in Calculus [on hold]

How to find the sum of the following infinite series assuming $|x|< 1$ ? $$5x^4+5x^5+5x^6+5x^7+⋯$$
-1
votes
2answers
49 views

How to determine the sum of the following infinite series problem? [on hold]

How to determine the sum of the following infinite series problem? $$\sum_{i=1}^\infty \frac{(-4)^{n-1}}{9^n}$$ I would get infinity for the following summation and it would diverge, but that isn't ...
0
votes
2answers
42 views

Does this sequence converge? Alternating and exponential

$$\sum_{k=1}^{\infty}\left(-1\right)^k\frac{\left(k+1\right)^{k+1}}{k^{k+2}}$$ I started to use Dirichlet's test. However, the latter half does not decrease to 0. I am unsure of what to do.
0
votes
3answers
27 views

How can I judge the series is convergence or divergence?

How can I judge the series is convergence or divergence? $$\sum_{n=1}^\infty {\Big({n}^{\frac{1}{n}}-1\Big)}^{n}$$ I don't know how to estimate ${n}^{\frac{1}{n}}-1$ when $n→∞$