Questions on the evaluation of limits.
96
votes
10answers
13k views
What is the result of infinity minus infinity?
What is $\infty - \infty$?
Is it $\infty$ or $0$ or what?
38
votes
2answers
2k views
Proof of 1=0 by mathematical induction?
I got stuck with a problem that pop up in my mind while learning limits. I am still a high school student.
$\lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{m})=0$
...
33
votes
10answers
1k views
What it the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?
What it the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$?
Using the definition:
$$\lim_{n\rightarrow\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$
I finally get 2 ...
30
votes
7answers
2k views
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim\limits_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my ...
28
votes
5answers
813 views
$\lim_{n\rightarrow\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
I'm supposed to calculate:
$$\lim_{n\rightarrow\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$
By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. ...
25
votes
2answers
486 views
Computing $ \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$
I would like to compute:
$$ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$$
I wanted to use Fubini's theorem for double series but $$ ...
23
votes
4answers
507 views
Limit of $\log (\log( … \log((n)^ {(n-1)^ {…}})))$
This is a spinoff of this question
Defining
$$f_0(x) = x$$
$$f_n(x) = \log(f_{(n-1)} (x)) \space (\forall n>0)$$
and
$$a_0 = 1$$
$$a_{n+1} = (n+1)^{a_n} \space (\forall n>0)$$
How to ...
22
votes
5answers
1k views
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$
In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} ...
22
votes
3answers
956 views
Why is this series of square root of twos equal $\pi$?
Wikipedia claims this but only cites an offline proof:
$$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$
for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
22
votes
8answers
1k views
When two functions are equal, but not.
I haven't looked into it much, but this is something I've been aware of that I know I need to look into.
When I have a function $f(x)=\frac{x+1}{x+1}$, There is a discontinuity at $x=-1$, yet ...
21
votes
6answers
936 views
Proofs of $\lim\limits_{n \to \infty} \left(H_n - 2^{-n} \sum\limits_{k=1}^n \binom{n}{k} H_k\right) = \log 2$
Let $H_n$ denote the $n$th harmonic number; i.e., $H_n = \sum\limits_{i=1}^n \frac{1}{i}$. I've got a couple of proofs of the following limiting expression, which I don't think is that well-known: ...
21
votes
2answers
469 views
Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$
I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating ...
21
votes
1answer
402 views
Convergence of the series $\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$.
Please determine whether the series $\displaystyle\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$ converges.
(Note: In Mathematica, the result tends to converge. Moreover, this is a problem mis-copied from ...
21
votes
1answer
595 views
Repeated Factorials and Repeated Square Rooting
I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ...
19
votes
8answers
522 views
Infinite powering by $ i$ [duplicate]
Find the value of:
$i^{i^{i^{i^{i^{i^{....\infty}}}}}}$
Simply infinite powering by i's and the limiting value.
Thank you for the help.
