All Questions
46
questions
1
vote
1
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116
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Upper bound of a sum of series
How can I find a tight upper bound for the following expression:
$\sum\limits_{i=1}^{k} a_i \sum\limits_{j = 1}^{i} \frac{1}{b_j} = a_1 \frac{1}{b_1} + a_2 (\frac{1}{b_1} + \frac{1}{b_2}) + \dots + ...
- 41
1
vote
1
answer
180
views
Prove that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\ge\frac{3}{2}$
For $a\geq b\geq c >0$. Prove that $$\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}\ge\dfrac{3}{2}$$
$a=100;b=1;c=1/100$ it's wrong ???
$\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}\ge\dfrac{3}{2}$ ...
- 1,898
2
votes
3
answers
152
views
Inequality related to sum of reciprocals: $\sum_{k=1}^{n} {\frac {1} {k^2}} > \frac {3n}{2n+1}$?
For every integer $n>1$, prove that :
$\sum_{k=1}^{n} {\frac {1} {k^2}} > \frac {3n}{2n+1}$
I don't seem to find any clue on how to relate the left side of the inequality to the right side.
I ...
user399078
6
votes
2
answers
144
views
How to show $\sum\limits_{r=0}^n \frac{1}{r!} \lt\left (1 + \frac{1}{n}\right)^{n+1}$ for all $n \ge 1$?
Using the binomial expansion, it is quite is easy to show that $$\left(1+\frac{1}{n}\right)^n \le \sum_{r=0}^{n} \frac{1}{r!} $$ for all $n\in\mathbb{Z^+}$, with equality holds when $n=1.$ (Can it be ...
3
votes
2
answers
144
views
Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
As the title says I need to prove the following by induction:
$$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$
When trying to prove that P(n+1) is true if P(n) is, then I ...
- 706
-4
votes
2
answers
110
views
proving $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ [closed]
how can one prove that:
$\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$
For all natural $n$,
without using induction?
thank you.
- 77
2
votes
2
answers
64
views
Alternate method to prove this series in a better way
Prove that $\frac{1.2 + 2.3 + 3.4 + .....+ n(n + 1)}{n(n + 3)} \ge \frac{n + 1}{4}$ for $n\ge1$
My attempt :
Breaking the series into two different series
$$ S_1 = \sum_{i = 0}^n i^2 = \frac{n(n + ...
- 1,801
7
votes
1
answer
244
views
Prove that $\frac{1}{1+a_1+a_1a_2}+\frac{1}{1+a_2+a_2a_3}+\cdots+\frac{1}{1+a_{n-1}+a_{n-1}a_n}+\frac{1}{1+a_n+a_na_1}>1.$
If $n > 3$ and $a_1,a_2,\ldots,a_n$ are positive real numbers with $a_1a_2\cdots a_n = 1$, prove that $$\dfrac{1}{1+a_1+a_1a_2}+\dfrac{1}{1+a_2+a_2a_3}+\cdots+\dfrac{1}{1+a_{n-1}+a_{n-1}a_n}+\dfrac{...
- 15.6k
0
votes
1
answer
139
views
Proof that the square root of the mean of the squares is always greater than or equal to the mean of weighted values
I couldn’t think of a better title, but basically you are given some values $x_1$, $\ldots$, $x_n$ and some weights $p_1$, $\ldots$, $p_n$ (with $x_k\in\mathbb{R}$ and $p_k\in[0,1]$, also $p_1+\ldots+...
- 338
2
votes
2
answers
129
views
Proving that $\sum_{i=1}^n\frac{1}{i^2}<2-\frac1n$ for $n>1$ by induction [duplicate]
Prove by induction that
$1 + \frac {1}{4} + \frac {1}{9} + ... +\frac {1}{n^2} < 2 - \frac{1}{n}$ for all $n>1$
I got up to using the inductive hypothesis to prove that $P(n+1)$ is true but I ...
- 832
1
vote
1
answer
147
views
Upper bound for $\sum_{i=2}^{n} 1/(i\log(i))$
I tried to find the asymtotic upper bounds for these summations
$$\sum_{i=2}^{n} 1/(i\log(i)) \text{ and } \sum_{i=2}^{n} 1/(i\log(i)\log\log(i)) .$$
My guess is that they might be bounded by $O(\log\...
- 783
12
votes
2
answers
937
views
$\sum\limits_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod \limits_{j=1}^nx_j}} \ge 1$, for all $x_i>0.$
Can you prove the following new inequality? I found it experimentally.
Prove that, for all $x_1,x_2,\ldots,x_n>0$, it holds that
$$\sum_{i=1}^n\frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod\limits _{j=...
- 2,040
5
votes
3
answers
450
views
Non-induction proof of $2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1$
Prove that $$2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1.$$
After playing around with the sum, I couldn't get anywhere so I proved inequalities by induction. I'm however ...
- 2,916
1
vote
1
answer
109
views
When $(\sum_{i=1}^nk_i < \prod_{i=n}^ni^{k_i}k_i!)$?
Consider $\Omega \subset \mathbb{N}$ a finite subset of $\mathbb{N}$, $\phi: \Omega \rightarrow \mathbb{N}$ an enumeration of $\Omega$ such that $\phi(\omega)=i$ and $|\Omega|=n$,
$$
\sum_{i=1}^n\...
- 4,899
10
votes
3
answers
718
views
Prove that $\sum\limits_{k=1}^n \frac{1}{k^2+3k+1}$ is bounded above by $\frac{13}{20}$
I want ask a question about a sum. The exercise is as follows:
Prove the following inequality for every $n \geq 1$:
$$\sum\limits_{k=1}^n \frac{1}{k^2+3k+1} \leq \frac{13}{20} .$$
- 101