# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

59,892 questions
Filter by
Sorted by
Tagged with
79 views

### Find the limit of $4^n a_n$, for the recurrent sequence $a_{n+1}=\frac{1-\sqrt{1-a_n}}{1+\sqrt{1+a_n}}$

Given the recurrence relation $$a_{n+1}=\frac{1-\sqrt{1-a_n}}{1+\sqrt{1+a_n}}$$ which is easy to find $$a_n\to0, \quad b_n=\frac{a_{n+1}}{a_n}\to\frac1{4}$$ hence $a_n\sim4^{-n}$, or with some ...
• 1,538
12 views

### Find $\alpha$ and $\beta$ such that $\sum_{n=1}^{\infty}\arctan({\frac{1}{n^{\alpha}}})-e^{\frac{2}{n^{\beta}}}+1$ converges

I want to find $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$ such that the following series converges: $$\sum_{n=1}^{\infty}\arctan({\frac{1}{n^{\alpha}}})-e^{\frac{2}{n^{\beta}}}+1$$ I have thought ...
• 343
14 views

### $\{x_n\},\{y_n\}$ two monotone sequences $\in \mathbb R$ such that $\sum x_n y_n$ converges. Which of the following is/are true? [closed]

a. At least one of $\{x_n\},\{y_n\}$ is bounded. b. $\{x_n\},\{y_n\}$ both are bounded. c. At least one of $\sum x_n,\sum y_n$ is convergent. d. $\sum x_n,\sum y_n$ both are convergent.
37 views

### How is this series diverging given this approxiamation?

I am given that a series follows the following formula: $$\sum_{n=1}^{\infty} 1/\sqrt{(n^2+n)} %$$ I approxiamate it with the following integral: $$\int_1^∞ 1/x \, dx = ln(∞)-ln(1)$$ Which simplifies ...
• 103
26 views

### Confusion to a solution to finding the Principal Part of a Laurent Series

Hello I am trying to find the Principal part at the pole $z = 1$ of the function $\frac{z}{(z^2-1)^2}$. Clearly, the function has a pole at $z=1,-1$, and by a theorem, $z=1$ is an order $2$ pole. Now ...
• 147
1 vote
59 views

### Find all $x \in \mathbb{R}$ such that $\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n}$ converges.

Find all $x \in \mathbb{R}$ such that $$\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n}$$ converges. My work: I tried to find for which $x$ partial sum $$\sum_{n=2}^{m} \sin(nx)$$ is bounded because, we ...
• 311
49 views

• 759
1 vote
43 views

• 999
20 views

### Is the limit of this sequence Lipschitz-continuous?

Suppose I have sequences $(f_{i,n})_n$ for $i=0,1,..,m$ of $M$-Lipschitz functions from $\mathbb{R}$ to $\mathbb{R}$. Each of those uniformly converges to an $M$-Lipschitz function $f_i$. Now ...
1 vote
31 views

### How do I prove that this sequence is increasing?

If $a_k = \sum_{i=0}^{k-1} \left( \frac{1}{a_k} \right)^i$ for $a_k \neq 0$, how do I show that $a_k\in\mathbb{R}$ (strictly) increases as $k\in\mathbb{N}$ increases? I've tried induction but I just ...
1 vote
36 views

### Is there $\sigma_\infty$ such that $\sigma_\infty$ is a subsequence of $\sigma_n$ for all $n \in \mathbb N$?

I have come across this "Diagonal method" in this lecture note. Theorem: Let $S$ be a non-empty set. For each $n\in \mathbb N$, let $\sigma_n$ be a sequence of elements of $S$. We denote by ...
• 10.2k
1 vote
22 views

• 2,122
34 views

### How the converging geometric series discovered or created? [closed]

It is easy to create new diverging geometric series. But how the converging geometric series evolve? Do they already exist in nature and waiting to be discovered or can we create them artificially?
• 173
1 vote
69 views

### Sum of n + n(n-1) + n(n-1)(n-2) + ... + n!

This is to work out the time complexity of a computer science problem (write an algorithm to calculate the permutations of an array of n distinct integers). Various answers on leetcode say the sum ...
1 vote
27 views

### Find the difference equation given the general solution $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$

Given that $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$, is the general solution to a difference equation, how do you work backwards to find the difference equation?
63 views

### How to compute $\displaystyle\prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}}$?

In my text book it is stated (without any explanation) that $$\prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}} = e^{e^x - 1}$$ and I can't really think of how one can show this.
• 33