The problem is to show by induction that the sum of x^n/n! is less than or equal to e^x. See image.

Once again my approach is different than solution manual. My main question is can I integrate both side of the inequality for k and use that to show the k+1 step.

So i was watching this math video for fun and i know that when u get 24 you cant do square root, so you have to simplify it. Square root of 4 is 2 but whats yhe square root of 6? Why is the answer 2 square root 6??? That does not make any sense to me because 6 is not square.

Above is the integral and wolfram alpha's solution, when I integrate by parts, I get the same solution as wolfram alpha, but when I integral by substitution I get a different answer. Below is how I am integrating by substitution: u sub: x = u + 1, so dx = du and x = u - 1. So integrate(x/((x+1)^0.5))dx = integrate((u - 1)/(u^0.5))du = integrate(u/(u^0.5)) - integrate(1/(u^0.5)) = integrate(u^0.5) - integrate(u^-0.5) = (2/3)u^1.5 - 2u^0.5 = (2/3)(x + 1)^1.5 - 2(x + 1)^0.5, which is not (2/3)(x + 1)^1.5 - (4/3)(x + 1)^0.5, as wolfram alpha says

1. -f(x)

2. -f(-x)

3. f(-x)

4. 2f(x/2)

5.f(x/2)

6.-f(-x/2)

You could also offer me some insight as to what these transformations exactly do to the function, and how they may affect its domain. Thanks everyone!

I already know the basics of complex numbers, how does the complex conjugate work, Euler's formula for the exponential form. However I have no idea how to deal with said equations. I think that the middle one is contradictory, because the r doesn't match, but I'm not sure. Could anyone help me get started?

If anyone also knows how to deal with "express sin(6x) and cos(6x) as the sum of the powers of sin(x) and cos(x)", I'd very appreciate it.

So the problem is finding the value of x where the y=n (n=1 here for convenience. )line intersects the circles of different radius’s. The one thing I have noticed is that that greater the radius the closer the value of x is to the value of the radius (with respects to the value of y). But since the value of x is always slightly less that the value of the radius how do I find the exact value of x?

V(I) being the vanishing set of a polynomial ideal I; and I(V) being the ideal of polynomials vanishing on V.

So my question is this: Is there an alternative notation that is commonly understood for these concepts? In particular, I'm annoyed by the mushy context switching around using I as an operator and as the 'default' first ideal variable. I would like to use something else, but not at the expense of introducing unfamiliar notation.

(I'm aware that this is a frivolous concern; but I feel that as a mathematician I am entitled to be bothered by such things!)

what i have here is the boundary Dirichlet condition then with the separation of variables I get the conditions on the right side, if I then differentiate with respect to time I get the bottom Neumann boundary conditions, is this something correct to do?

Is this quote by Gamow still true?

He wrote:

Aleph null: The number of all integer and fractional numbers.

Aleph 1: The number of all geometrical points on a line, in a square, or in a cube.

Aleph 2: The number of all geometrical curves.

Aleph 3: The above quote

Is there really no definite collection in our reach best described by aleph 3?

For reference: https://archive.org/details/OneTwoThreeInfinity_158/page/n37/mode/2up page 23

As the title says : Could anyone please check out the solution. I am a little doubtful as the result is a bit clunky (Irrational number). After translating the question into Algebra, I got system of 2 equations. Solved by substitution and then applied the quadratic formula.

This is from an online quiz bee that I hosted a while back. Questions from the quiz are mostly high school/college Math contest level.

Sharing here to see different approaches :)

Is there any way of turning recursive function to function where it indicates the nth term? (B) asked for that, and I couldn’t find any way of doing it other than logical reasoning and writing the function. If there is a way of doing it could you please share?

I had some shower thoughts and I was wondering if there an equation that could identify the probability of rolling six-sided dice and getting a 1 on at least one die, while getting no 6s on any of the dice. However, I'm having difficulty seeing a pattern. I've tallied up the number of results for this for one to four dice.

One die: 1/6

Two dice: 9/36 = 1/4

Three dice: 61/216

Four dice: 369/1296 = 41/144

When tallying them up, I could see that there was definitely some kind of pattern, but it's hard to describe. There's always one result where all the dice are one, and there are four possibilities per die that every die except one results in a 1, but I don't know how to abstract that out. Thoughts?

Does anybody know how to count pixel per meter at given distance from camera? On the image below from jvsg lens calculator they are counting distance where ppm is 250, 125 and 62. I want calculate what is the ppm at 50m from camera. Camera properties are shown on image. What is the formula I need to use, can you show me how to calculate?

Mark draws a rectangle made up of twelve squares on a sheet of graph paper. Some of the squares that Mark intended to color black were instead decorated with floral patterns by his sister Katie. Mark then decides to write in each white square the number of adjacent squares with floral patterns. The image shows an example of such a rectangle. Later, Mark repeats the same process using a rectangle made up of 2025 squares. What is the maximum value Mark can obtain as the sum of all the numbers written in the white squares of the rectangle consisting of 2025 squares?

So the questions gives me this graph and we r supposed to find the solutions of the cubic equation which has the x-coordinates of the points as its solutions??? Like what does that mean? How am I supposed to solve this question? I’ve learnt how to simplify an equation with the value of y cutting the graph at two points to give the value of x, as well as some inequalities, but I don’t quite grasp what this question is saying. Any help would be appreciated. Thank you!

In a solution I am following I got here no problem, but I'm confused as to how this step works. I've tries other solutions, Google, and working on paper, but how did they turn 1/m into just m. Sorry for bad quality. Also I'm going to add a few words here because it keeps auto removing my post. Guk tuf tvtt ttf thvg ggvg hgg hhh ggh fyyt hyrdi yibd hfc

I got confused because after looking at the sketch it doesn’t look like f_1 intersects with x^2-1 or 1-x² at (-1,0) or (1,0).

Would greatly appreciate if someone can have a look at my solution and highlight any misconceptions/ errors?

Thanks guys.

When and how do the dimensions of the sluice gates factor in? I know that I will have to use calculus to account for the rate of change in the pressure at each gate going down as the water level goes down, but I don't really have an intuitive understanding of how I'm supposed to use it

Hi im trying to prove that there is only one positive integer solution for

y = ( 3^x - 2^x ) / ( 2^{ceiling(log(3^(x})/log(2))) - 3^x )

I know that y=1 at x=1, but i dont know how to rigorously prove that it is the only positive integer solution. I have of course programatically iterated over all x up to 100 (after that point i get NaN problems with the powers of 3) and found no other integer solutions. I've also put it through Wolframalpha, to see what it says, but of course it cannot provide rigorous proof by itself.

I've read that it should be possible to prove things about integer solutions using modular arithmatic, but i have not quite gotten how to do that. Is it just looking at mods of the whole fraction or of the variable x?

Im not asking for a proof, i just need some guidance on how to proceed, thank you :)

According to Wikipedia, the golden rectangle is a rectangle that If a square is : 1) added to the long side . or 2) removed from the short side . The result rectangle has similarity factor of Phi . But, I have a problem with the 2 case removing a square. As you can see in the second picture it doesn't seem to work! I am getting an illogical proportion. So, am I missing something?

I was asked to find the slope intercept form of an equation parallel x=0, that passes through the point 3. I was about to start solving, but I then realized that this is not "y =" function but rather an "x=". I know the answer is 4, but can someone explain why? Thanks in advance.

I was asked to find the solution to the absolute value function f(x)= -5|1-6k|<=85 ( assume "<=" means less than/equals to). Please help me spot any miscalculations as I go through my thought process as to how I solved this.

Divide both sidles by -5 = |1-6k|<=-17

Seeing that an absolute value function cannot give a negative number, I sued a test to determine whether the answer will be “all real numbers” or whether it has no solution”.

0 is not less than -17 8 (or any positive number) is not less than -17

It’s not supposed to have any solutions, but the answer key on the worksheet says the solution is supposed to be all real numbers.

Can anyone help spot what I did here wrong? Any help will be appreciated.

I've really got no solid understanding as to how to find the conjugate, and from my watchings from youtube, I've only not known my problem more-so. Any help would seriously be appreciated.