All Questions
Tagged with sequences-and-series inequality
1,014
questions
5
votes
1
answer
197
views
Finding the integer part of a sum
I want to find the integer part of $$\sum_{n=1}^{10^9}\frac{1}{n^{2/3}}=S$$
I know there is a way using integration but I tried using a different approach.
I saw this approach with square roots but I ...
- 107
2
votes
1
answer
137
views
Order between the solutions of linear recursions [closed]
We consider the following 4 linear recursions:
$$
\begin{array}{llll}
u_{n+3}& = \frac 12 u_{n+2} & + \frac 14 u_{n+1} & + \frac 18 u_n \\
v_{n+3} & = \frac 12 v_{n+2} &+ \frac ...
- 1,363
5
votes
0
answers
173
views
Prove or disprove the limit of a sequence is negative.
I have a sequence of positive numbers $\{f_k\}$ such that all the odd terms sum up to 1 and so do all the even terms, i.e. $\sum_{k=1}^{\infty}f_{2k-1}=\sum_{k=1}^{\infty}f_{2k}=1$, and $1>\sum_{i=...
- 165
2
votes
0
answers
53
views
Proof explanation: how to see that $(u_n)$ is bounded above by $v_0$
I am new to this and I am trying understand this proof of Monotone-sequences property $⇒$ least upper bound property.
Let $A$ be a non-empty set that's bounded above. Pick $u_0, v_0$ such that $u_0$ ...
- 95
1
vote
1
answer
95
views
Non negativity involving sequences
Define for $n\in\mathbb{N}$ $$a_n=\left[n\sum_{k=1}^{n}\frac{1}{k^5}\right]$$ where $[x]$ denotes the greatest integer $\leq x$.
Prove that $$(n+1)a_n-n a_{n+1}+1\geq 0 \ \ \forall n\geq 1$$
$$(n+1)...
- 928
-2
votes
1
answer
98
views
Convergence of $\sum^{\infty }_{n=0} |a_{n}-b_{n}|$ from convergence of $\sum^{\infty }_{n=0} |a_{n}|$ and $\sum^{\infty }_{n=0} |b_{n}|$
Exercise 12.1.15 of Tao's Analysis II book asks the reader to show that the function $d:X\times X\rightarrow \mathbb{R} \cup \left\{ \infty \right\} $ defined by $d\left( a_{n},b_{n}\right) =\sum^{\...
1
vote
1
answer
56
views
Upper bound for $ \prod_{i=0}^{k^2} \left(1 + \frac{n-1}{k+(n-1)i}\right) \left(1 - \frac{1}{k+(n-1)i}\right)^{n-1}$
I am working on finding a good upper bound for the following product:
$ \prod_{i=0}^{k^2} \left(1 + \frac{n-1}{k+(n-1)i}\right) \left(1 - \frac{1}{k+(n-1)i}\right)^{n-1} $
With the following ...
- 85
2
votes
2
answers
99
views
Prove that $\exp \left(\dfrac{-2 \sum_{n=0}^{K-1} \frac{2^n}{n!}}{ \sum_{n=0}^K \frac{2^n}{n!}}\right) \sum_{n=0}^{K-1} \dfrac{2^n}{n!} -1 \geq 0$
I want to show the following inequality:
$$\exp \left(\dfrac{-2 \displaystyle \sum_{n=0}^{K-1} \frac{2^n}{n!}}{\displaystyle \sum_{n=0}^K \frac{2^n}{n!}}\right) \displaystyle \sum_{n=0}^{K-1} \dfrac{2^...
user1325284
3
votes
1
answer
107
views
Conditions that a sequence should satisfy to be an eventually monotone sequence
Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that:
$a_n\in[0,1]$, $\forall n\in\mathbb{N}$
$\lim_{n\to\infty}a_n = 0$
$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = 1$
$a_{n+1} \leq a_n$...
- 888
1
vote
0
answers
29
views
Quotient of decreasing bounded sequences is definely increasing
Suppose that $\{a_k\}_{k\in\mathbb{N}}$ is a bounded sequence of real numbers with $a_k\in[0,1]$, $\forall k\in\mathbb{N}$. Suppose that $a_k$ is decreasing ($a_k > a_{k+1})$ and that $\lim_{k\to\...
- 888
1
vote
6
answers
101
views
Assuming the conclusion while trying to prove an inequality
I am trying to prove the following inequality:
$$ \ln{\frac{2n} {n+1}} \lt \ln{\frac{2n+2} {n+2}} $$ for any $$n \gt 0$$
I started with the above statement, took antilog on both sides and rearranged ...
- 13
2
votes
2
answers
78
views
Minimum and maximum sum of numbers and sum of squares
If $x_1,x_2,...,x_n,...x_{2n}$ be non negative numbers such that
$x_1+x_2+...+x_n+...+x_{2n}=1$,
we need to find the minimum and maximum value of
$\sum_{i=1}^nx_i$ +$\sum_{j=n+1}^{2n}x_j^2$.
As $x_j^2\...
- 23
1
vote
0
answers
40
views
Equivalence in $l_q$ spaces
Let $\theta \in (0,1)$ and $1\leq q<\infty$. Let $\lambda=(\lambda_n)$ a sequence of real (or complex) numbers such that $$||(2^{-n\theta}\lambda_n)||_q<\infty.$$ Now, let $\beta=(\beta_n)$ ...
6
votes
1
answer
245
views
A question on proving an inequality involving a sequence of real numbers
Let $a_n$ be a sequence of real numbers such that $1=a_1 \le a_2 \le a_3 \le \cdots \le a_n.$
Additionally, we have that $a_{i+1}-a_i \le \sqrt{a_i},$ for all $1 \le i <n.$
Then prove that $$\sum_{...
- 779
1
vote
1
answer
45
views
Let $f_n:[0,1] \to \mathbb {R}$ with $f^{'}_n$ uniformly bounded by $1$ and exists $f$ s.t $f_n \to f$ pointwise. Prove convergence is uniform.
Question:
Let $f_n:[0,1] \to \mathbb {R}$ with $f^{'}_n$ uniformly bounded by $1$. Let $f:[0,1] \to \mathbb R$ s.t $f_n \to f$ pointwise in $[0,1]$. Prove convergence is uniform.
My attempt:
So for ...
- 515