# Tagged Questions

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### Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
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### Calculate exact value of and infinite sum [duplicate]

Im trying to find the exact value of the infinite sum : 3 + 1/3 + 1/27 + 1/243 + 1/2187 + ... I can see that to generate new terms we take the previous term and divide by 9 or multiply by 9. Not ...
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### Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
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### Infinity and Hilbert's hotel paradox

I did some infinite series calculations while studying Fourier analysis and the concept of infinity really bugs me. I haven't read or heard not one sensible explanation yet (for me), what infinity ...
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### Naive calculations with infinite series [duplicate]

In the realm, where the sum of natural numbers is $-1/12$ : $1+2+3+4+...=-1/12$ Is this true?: $2+4+6+8+...=2*(1+2+3+4+...)=-2/12$ Can this kind of naive calculations always be done? -or are there ...
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### How to show by the Root Test that $\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$ converges or diverges

How do I show by the Root Test that $$\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$$ converges or diverges? This is what I have done so far. Since we take $\sum\limits_{i=1}^\infty \sqrt[n]{|a_n|}$, we ...
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### Limits of infinite processes that terminate in finite time - checking my understanding?

I am a computer scientist by training, but have a fair amount of math background that I've picked up through classes, teaching, and general interest. A student of mine posed a question to me. I think ...
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### What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series (An = 1 + 1/2 + ... + 1/n) and this natural logarithm L = ...
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### Are the integers closed under addition… really?

Okay so I'm a 3rd year undergraduate studying Mathematics. I've proved in group theory countless times that the integers are closed under addition. It's obvious to me that they are. However this has ...
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### Is there any mathematical or physical situations that $1+2+3+\ldots\infty=-\frac{1}{12}$ shows itself? [duplicate]

I just saw the proof that $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts. I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, ...
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### An intuitive reasoning for 1+2+3+4+5… + ∞ = -1/12? [duplicate]

I was just watching this video: http://www.youtube.com/watch?v=w-I6XTVZXww In it, a professor working at the Nottingham University( Dr. Ed Copeland I think) shows how 1+2+3+4+5....+ ∞ = -1/12 Is this ...
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### How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
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### Laurent Seies and Res

Prove that for any Laurent series f(t) one has "Res(f') = 0"? I know for a Laurent series of a complex function f is a representation of that function as a power series which includes terms of ...
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### Limit for $\lim _{n \to \infty}(n+2)^{2}\sin\frac{1}{n}$

Can't prove the limit $$\lim_{n \to \infty}(n+2)^{2}\sin\frac{1}{n}=\infty.$$ by definition it should start: Let $M>0$. There exists an $N>0$ for every $n>N$: ...
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### Proving the product of two series diverges to infinity.

Proving the product of two series diverges to infinity, given that one series (An) converges to a limit L and (Bn) diverges to infinity, I have to prove that the product of the two series (AnBn) ...
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### Hilarious Comic … DiffyQ and infinity ensue…

I ran across this comic, and it's gold. It is orginially published here If I am correct, the first panel alone defines a self-referential loop if not a differential Equation: $X$: Amount of Black ...
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### Difference between approaching and being exactly a number

When we take a limit, we say that the value is never equals that number, but approaches it, like in $$\lim_{n\to\infty}\frac{1}{n} = 0.$$ It never reaches $0$, but becomes closer and closer to $0$. ...
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### $\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$ Are there any known ...
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### Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
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### Are the number of terms in an infinite series even or odd?

This question arose after I saw a youtube-vid where Grandi's series was discussed.It seems that the sum of the series will be 0 for an even, and 1 for an odd number of terms, where a term is defined ...
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### Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
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### Absolute value of infinite sum smaller than infinite sum of absolute values

A question emerging from an exercise in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press. The exercise consists in showing that if $\sum_{i=1}^\infty x_i$ ...
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### A question with infinity [Part 2]

If you haven't seen my first post regarding infinity you can find it here: A question with infinity Thanks for all the constructive comments on my first post, and creative answers to my questions. ...
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### 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 ...
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### Why do we say the harmonic series is divergent? [duplicate]

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 ...
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### Evaluation of $\sum_{x=1}^{\infty}x^{-x}$ [duplicate]

Possible Duplicate: “Closed” form for $\sum \frac{1}{n^n}$ Is it possible to evaluate this sum, and if so, how would you do it? This question has been irritating me for a ...
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### 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 ...
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### Sierpinksi like triangle construction. How to find the number of triangles in each iteration?

So here is the question: If we look at the Sierpinski triangle (left column of attached image) and think about how many triangle's it takes to make the shape at each iteration we can get the sequence ...