Questions on the evaluation of limits.
99
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 ...
32
votes
8answers
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
874 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
492 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 $$ ...
23
votes
5answers
1k views
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$
In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} ...
23
votes
3answers
978 views
Why is this series of square root of twos equal $\pi$?
Wikipedia claims this but only cites an offline proof:
$$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$
for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
23
votes
4answers
510 views
Limit of $\log (\log( … \log((n)^ {(n-1)^ {…}})))$
This is a spinoff of this question
Defining
$$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)$$
How to ...
22
votes
8answers
1k views
When two functions are equal, but not.
I haven't looked into it much, but this is something I've been aware of that I know I need to look into.
When I have a function $f(x)=\frac{x+1}{x+1}$, There is a discontinuity at $x=-1$, yet ...
21
votes
6answers
947 views
Proofs of $\lim\limits_{n \to \infty} \left(H_n - 2^{-n} \sum\limits_{k=1}^n \binom{n}{k} H_k\right) = \log 2$
Let $H_n$ denote the $n$th harmonic number; i.e., $H_n = \sum\limits_{i=1}^n \frac{1}{i}$. I've got a couple of proofs of the following limiting expression, which I don't think is that well-known: ...
21
votes
3answers
321 views
$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $
What would you suggest here?
$$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $$
21
votes
2answers
477 views
Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$
I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating ...
21
votes
1answer
418 views
Convergence of the series $\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$.
Please determine whether the series $\displaystyle\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$ converges.
(Note: In Mathematica, the result tends to converge. Moreover, this is a problem mis-copied from ...
21
votes
1answer
598 views
Repeated Factorials and Repeated Square Rooting
I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ...
20
votes
6answers
800 views
Uses of $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$
I have been wondering whether the following limit is being used somehow, as a variation of the derivative:
$$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$
Edit:
I know that this limit is defined in ...
20
votes
2answers
449 views
Crafty solutions to the following limit
The following problem came up at dinner, I know some ways to solve it but they are quite ugly and as some wise man said: There is no place in the world for ugly mathematics.
These methods are using ...
19
votes
8answers
536 views
Infinite powering by $ i$ [duplicate]
Find the value of:
$i^{i^{i^{i^{i^{i^{....\infty}}}}}}$
Simply infinite powering by i's and the limiting value.
Thank you for the help.
18
votes
3answers
1k views
Find the limit $\lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right)$
I'd love your help with finding the following limit:
$$\lim_{n\to \infty }\cos (\pi\sqrt{n^{2}-n}).$$
I was asked to find this limit, but honestly I believe that it doesn't exist.
According to Heine ...
18
votes
4answers
675 views
Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$
Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.$$
This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ...
18
votes
4answers
359 views
Limit of series involving ratio of two factorials
$$
\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}
$$
The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
18
votes
3answers
377 views
When L'Hôpital's Rule Fails
I was discussing L'Hôpital's Rule with a Calculus I student earlier today. I mentioned that if the limit from LH doesn't exist, then L'Hôpital's Rule tells us nothing about the original Limit.
A ...
17
votes
7answers
997 views
Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?
Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$
I.e - does the second half of the harmonic series go to zero?
I know that for a finite number of terms the limit of the sum is ...
17
votes
3answers
818 views
$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$ Evaluating this limit
Evaluate $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$
17
votes
3answers
499 views
How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$?
This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this.
...
17
votes
1answer
451 views
A continued fraction involving prime numbers
What is the limit of the continued fraction
$$\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{11+\cfrac{1}{13+\cdots}}}}}}\ ?$$
Is the limit algebraic, or expressible in terms of e or ...
17
votes
3answers
434 views
Why can't $\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}>2$?
So we have$$\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}=x\\\sqrt{2}^x=x$$where $x=2$ heuristically seems like a good solution. However, $x=4$ seems like an equally good solution. I was told in passing ...
17
votes
3answers
336 views
Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$
I wonder what kind of functions satisfy
$$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$
I suppose all functions must be continuous.
16
votes
4answers
641 views
Wrong Wolfram|Alpha limit?
I have this function:
$$ f(x,y) = \frac {xy}{|x|+|y|} $$
And I want to evaluate it's limit when $$ (x,y) \to (0,0)$$ My guess is that it tends to zero. So, by definition, if:
$$
\forall \varepsilon ...
16
votes
4answers
420 views
Computing $\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$
I got stuck at the following problem.
Let $f\in C([0,\pi/2])$, then compute
$$
\lim_{n\to+\infty}n\int\limits_{0}^{\pi/2}xf(x)\cos ^n xdx
$$
Could you suggest a helpful idea?
16
votes
4answers
1k views
How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?
I was trying to work out a problem I found online. Here is the problem statement:
Let $f(x)$ be continuously differentiable on $(0, \infty)$ and suppose $\lim\limits_{x \to \infty} f'(x) = 0$. ...
16
votes
2answers
363 views
Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin…(\sin(x)..)$
I was asked today by a friend to calculate a limit and I am having
trouble with the question.
Denote $\sin_{1}:=\sin$ and for $n>1$ define $\sin_{n}=\sin(\sin_{n-1})$.
Calculate ...
16
votes
5answers
373 views
limit of : $a_{n+2} =\frac{1}{a_n} + \frac{1}{a_{n+1}}$
$a_n$ is a real sequence, $a_1,a_2$ are positive and for all $n>2$ : $$ a_{n+2} =\frac{1}{a_{n+1}} + \frac{1}{a_{n}}.$$
Prove that: $\displaystyle \lim_{n\to \infty} a_n$ exists, then find it.
...
16
votes
3answers
357 views
An interesting sum to infinity
Is there any simple way of computing the following sum?
$$\sum_{k=1}^\infty \frac1{k\space k!}$$
16
votes
3answers
377 views
The limit of $\frac{n+\sqrt{n}+\cdots+\sqrt[n]{n}}{n}$
How do I compute the following limit or show it doesn't exists?
$$\lim_{n\rightarrow\infty}\frac{n+\sqrt{n}+\cdots+\sqrt[n]{n}}{n}$$
I've struggled with this problem for a while now so I would ...
16
votes
2answers
370 views
Is there a step by step checklist to check if a multivariable limit exists and find its value?
Do we rely on certain intuition or is there an unofficial general crude checklist I should follow?
I had a friend telling me that if the sum of the powers on the numerator is smaller then the ...
16
votes
1answer
322 views
Three sequences and a limit(own)
Let us consider three sequences $(a_n)_{n\ge1}$, $(b_n)_{n\ge1}$ and $(c_n)_{n\ge1}$ having the properties:
$a_{n},\ b_{n},\ c_{n}\in\left(0,\ \infty\right)$
...
16
votes
2answers
333 views
Limit of sum with binomial coefficient
Prove that:
$$\lim_{n\to +\infty}\sum_{k=0}^{n}(-1)^k\sqrt{{n\choose k}}=0$$
I completely don't know how to approach. Is it very difficult?
15
votes
11answers
2k views
$ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
15
votes
3answers
1k views
On applying the quadratic formula to a first-degree equation
You're probably thinking, "Why?" Please let me explain...
It is (very) well-known that
$$ \forall (a,b,c,x) \in \mathbb{C}^* \times \mathbb{C}^3: ax^2 + bx + c = 0 \Leftrightarrow x = \frac{-b \pm ...
15
votes
4answers
600 views
Finding $\lim\limits_{n \to \infty}{\frac{1^1+2^2+3^3+\cdots+n^n}{n^n}}$
$$\lim_{n \to \infty}{\frac{1^1+2^2+3^3+\cdots+n^n}{n^n}}.$$
With a first look this must give $1$ as a result but have a problem to explain it.
How can I do it?
Edit
I noticed that it is ...
15
votes
3answers
529 views
What kind of “mathematical object” are limits?
When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly ...
15
votes
6answers
863 views
Limits of $f(x)=x-x$
It's obvious that $f(x)=x-x=0$. But what exactly happens here?
You have a function $f(x)=x-x$ and you have to calculate the limits when $x\to \infty$
This'll be like this:
$$\lim\limits_{x\to ...
15
votes
2answers
360 views
Compute: $\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}} $
Compute the following limit:
$$\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}} $$
I'm interested in almost any approaching way for this limit. ...
15
votes
1answer
900 views
What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
I tried and got this
$$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$
$$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$
where $m$ is an ...
15
votes
4answers
4k views
When can you switch the order of limits?
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to ...
15
votes
1answer
228 views
The Fibonacci sum $\sum_{n=0}^\infty \frac{1}{F_{2^n}}$ generalized
The evaluation,
$$\sum_{n=0}^\infty \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}=\left(\frac{1-\sqrt{5}}{2}\right)^3+\left(\frac{1+\sqrt{5}}{2}\right)^2$$
was recently asked in a post by Chris here.
I ...
15
votes
2answers
234 views
How to evaluate $\sum\limits_{n=1}^{+\infty}\frac{1}{1^k+2^k+\cdots+n^k}$?
There is post here asking for $$\sum_{n=1}^{+\infty}\frac{1}{1^2+2^2+\cdots+n^2}.$$
And the answer is $18-24\ln 2$.
It is easy to elvaluate that ...
14
votes
5answers
1k views
Which infinity is meant in limits?
For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why?
14
votes
10answers
1k views
Why isn't $\lim_{x\to\infty}(1+\frac{1}{x})^{x}= 1$?
Given $\lim_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim_{x\to\infty}(1+0)^{x}$, making the result "$1$"? Obviously, it's wrong, as the true value is $e$. Is it because the ...
