Tagged Questions
1
vote
1answer
71 views
infinity times infinitesimal - what happens?
So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration.
Also, can we divide infinite number by ...
0
votes
2answers
55 views
Understanding limits at infinity with regard to the definition of a limit
This is sort of a follow up to my previous question
Say you have
$$
\lim_{x\to +\infty} f(x)
$$
where $f : \mathbb{R} \to \mathbb{R} , x \in \mathbb{R}$
What exactly does this mean? From the last ...
2
votes
1answer
87 views
Infinity = Undefined?
Let's start with the equation $y = |1/(x-1)|$. The positive and negative limit of $x$ at $1$ both approach $+∞$, but at $x = 1$, $y$ is undefined.
I know this is because the denominator of the ...
0
votes
1answer
42 views
Calculus Limit -> inf Question. Kindly Explain the First Step, encircled in red color.
Link to view Solution of my question in image format:
Solution is correct.
Kindly Explain the First Step, encircled in red color.
0
votes
1answer
39 views
Comparing Improper Integrals Involving Infinity
From my current understanding: $K>J$ and $L>K$ , therefore $L>K>J$. How can I compare the first integral $I$ ?
0
votes
1answer
181 views
Comparison Theorem for Integral Calculus
I have narrowed it down to C, E, and F, since we know that $1/x^{1/5}$ is always greater than the original function for all $x\geq 1$. However, the second set of conditions is more difficult to ...
0
votes
1answer
56 views
Identifying an Error in Determining the Convergency of an Infinite Series
Given the infinite series of $(-1)^n/(nln(n))$ for $n = 2,3,4,\ldots$ to infinity, is the series conditionally convergent, absoultely convergent, or divergent?
I took two approaches to solve this ...
1
vote
0answers
65 views
Simplifying this infinite series [duplicate]
Possible Duplicate:
How can I evaluate $\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$
I have an infinite series like so:
$$\sum_{i=0}^\infty (i+1)x^i$$
or basically
$$ 1 + 2x + 3x^2 + 4x^3 +... ...
3
votes
4answers
130 views
Limit of difference of two irrational functions
Firstly, this is not a homework. I just want to solve this limit for my own curiosity and self-learning. I have tried to solve this limit for 5-6 hours with no luck. Then I tried to read information ...
1
vote
1answer
3k views
Whats infinity divided by infinity?
This should be a simple question but i just want to make sure.
I know from infinity/infinity is undefined.
However if we have 2 equal infinities divided by each other it would be 1?
And if we have ...
1
vote
7answers
679 views
Why do we say the harmonic series is divergent?
If we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...
1
vote
4answers
152 views
why $\lim_{x\to-\infty}(\sin x+2)\ln(-x)=\infty$?
Why does $\lim_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$?
Breaking up the limit:
$\lim_{x\to-\infty}(\sin x+2)$ DNE because it oscillates between 1 and 3
$\lim_{x\to-\infty}\ln(-x) = \infty$
...
2
votes
2answers
90 views
Looking for some function such that $\lim\limits_{x\to\infty}f(x) \ne \infty$
I am looking for a function $f$ that is differentiable and $f'(x) \ge c \gt 0$ for all $x \in \mathbb{R}$ and $\lim\limits_{x\to\infty}f(x) \ne \infty$?
Is there such function, or am I wasting my ...
3
votes
1answer
208 views
Why do some divergent series apparently seem to converge (e.g. Grandi's series)?
Grandi's series is defined as:
$$\sum_{n=0}^{\infty} (-1)^n = 1 - 1+1-1+\cdots$$
By plainly looking at this series it seems like the value of it is either $1$ or $0$ by doing the following ...
6
votes
1answer
263 views
Am I thinking about infinitesimals correctly?
I was all set to begin Calc 2 when I thought, "I should have a more intuitive sense of what's happening with differentials before I move on."
I want to tell you what I've learned and ask you all to ...
12
votes
3answers
572 views
The Aleph numbers and infinity in calculus.
I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus?
Are they interchangeable/transposable in ...
3
votes
3answers
134 views
Continuity of $f(x)$ involving infinity
$f(x)= \frac{\sin(\pi x)}{x(1-x)}$
How can I define $f(0)$ and $f(1)$ to make $f(x)$ continuous on $[0,1]$?
I've found that the limit at $0 = \pi$, and the limit from the left at $1 = \infty$.
I ...
5
votes
6answers
520 views
Why is $\infty^0$ indeterminate?
In a recent test question I was required to us L'Hopital's rule to evaluate:
$$\lim_{x\to 0^+} x\ln{(e^{2x}-1)}$$
I assumed that anything multiplied by 0 would give an answer of 0. This turns out ...
1
vote
3answers
195 views
Surface under $\frac{1}{x}$ is $\infty$, while surface under $\frac{1}{x^2}$ is $1$?
Since the antiderivative of $\frac{1}{x}$ is $\ln(|x|)$, the surface under the graph of $\frac{1}{x}$ with $x>1$ is $\infty$.
However, the antiderivative of $\frac{1}{x^2}$ is $-\frac{1}{x}$, so ...
3
votes
2answers
55 views
interval for a product to infinity
I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, ...
5
votes
1answer
712 views
Limit approaches infinity on one side and negative infinity on other side
I know this is a simple question for most of you, but I am currently studying for a Calculus exam and was just wondering why an online calculator I am using to double-check my work was disagreeing ...
24
votes
4answers
2k views
How can a structure have infinite length and infinite surface area, but have finite volume?
Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.
One Dimension - Clearly this structure is infinitely long.
Two Dimensions - Surface Area = ...
