I am reading the following passage in these lecture notes (chapter 10, in the proof of theorem 10.3) on power series (and have seen similar statements in other texts):
I'm confused about ##|x_0|<R##.
If ##M=\sup (A)##, then for every ##M'<M##, there exists an ##x\in A## such that ##x>M'##...
The problem itself is easy. My question is regarding the proper use of inequality symbols.
I only need to do the first part to show where I am having the issue.
The forces I need to consider are the coaster car's weight ##W = mg## and the reaction ##A## of the tracks acting on it. With the...
Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
The assignment says proof by induction is possible, I cannot figure out how this is supposed to work out. Does somebody know the name of this by any chance? Seeing a derivation might help come up with an idea for a proof. Thank you everybody.
When I learned about Catalan's conjecture, I morphed the expression ##3^2-2^3## to ##a^b+b^a##. Then, out of curiosity, I compared ##a^b+b^a## to ##a^a+b^b##. I have tried many numbers (greater that or equal to 1) for both ##a## and ##b##, noticing that ##a^a+b^b## is bigger than ##a^b+b^a##. I...
Let’s say we are given two inequalities $$ x \lt y \\
a \lt b$$ then we can write (we can even prove it using logarithms) $$ ax \lt by$$ given that every number is positive.
In this article (Fact 4.1, point (ii) ) it is given that if ##x, y## are positive numbers and...
Homework Statement
I'm currently working through Spivak independently and have reached the problems at the end of ch. 1.
The problem is:
Prove that if 0 < a < b , then a < \sqrt{ab} < \frac{a+b}{2} < b
Homework Equations
Spivak's properties P1 - P12
The Attempt at a Solution
I was...
Homework Statement
Prove that ##\forall n \in \mathbb{N}##
$$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$
Homework Equations
Peano axioms and field axioms for real numbers.
The Attempt at a Solution
Okay so my first assumption was that this part...
Homework Statement
Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set.
Homework Equations
n/a
The Attempt at a Solution
Let ##x \in T##, then ##\exists \sigma \in S## such that ##x...
So, I know that the inequality √f(x)<g(x) is equivalent to f(x)≥0 ∧ g(x)> 0 ∧ f(x)<(g(x))^2. However, why does g(x) have to be greater and not greater or equal to zero? Is it because for some x, f(x) = g(x)=0, and then > wouldn't hold? Doesn't f(x)<(g(x))^2 make sure that f(x) will not be...
Homework Statement
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2. The attempt at a solution
I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete...
Homework Statement
[/B]
In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ?
2. Homework Equations The Attempt at a Solution
An absolute value property is
$$\lvert a \rvert \geq b \iff a\leq-b \quad \text{ or } \quad a\geq b,$$ for ##b>0##.
Is this true for the case ##a=0##?
I mean if ##a=0, \lvert a \rvert =0## so ##0 \geq b##. But ##b## is supposed to be ##b>0##, so we have a contradiction.
How can this property...
Homework Statement
Given : (y+2)(y-3) <= 0Homework EquationsThe Attempt at a Solution
Now, I have y-3 <= 0 or y+2 <= 0
Hence, y <= 3 or y <= -2
But how
is correct?
I think
is wrong because y <= -2.
Can someone please clarify?
Homework Statement
Let S denote (x,y,z) in R3 which satisfies the following inequalities:
-2x+y+z <= 4
x-2y+z <= 1
2x+2y-z <= 5
x >=1
y >=2
z >= 3
Homework Equations
How to find the dimension of the set S ?
The Attempt at a Solution
I have tried to transform the inequalities into matrix form...
I am trying to solve this inequality without using a factor table.
The problem
$$ \frac{x+4}{x-1} > 0 $$
The attempt at a solution
As I can see ##x \neq 1##. I want to muliply both sides of the expression with x-1 to get rid of it, from the fraction. But before that, I have to consider two...
Homework Statement
Homework Equations
With the regards to posting such a incomplete equation, I will soon put in the updated one
Thank you
The Attempt at a Solution
visual graph... didn't help
Homework Statement
Hi Guys,
This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help.
"E1. Starting with a point S (a, b) of the plane with 0 < b < a...
Hi!
At university I have got a problem set with lots of inequalities. Unfortunately there are no explanations given how to do them. In Highschool we only did very easy inequalities.
Therefore I am looking for a resource for inequalities. Especially for more difficult inequalities like $$ 1...
Homework Statement
What values of X between 0 and 2 pie radians satisfy each of the following:
1. |sinX|<0.5
2. |cosX|>0.5
Homework Equations
The Attempt at a Solution
Well the values of X lie between
1. -0.5 < sinX <0.5
2. cosX< -0.5 and cosX>0.5
How do you find the...
Find where cosθ = -0.8660, where (0 degrees ≤ θ < 540 degrees)
I am not sure how to solve this problem
I did: cosθ + 0.8660 = 0
Then I am not sure what to do
Hi, can someone help me with the following question? I don't know how to approach the proof :confused:
If a < b and c < d then ac < bd is true, supply a proof.
Thanks!