# All Questions

7k views

### Does .99999… = 1?

I'm told by smart people that 0.999... = 1 and I believe them but is there a proof that explains why?
3k views

### Value of $\sum\limits_n x^n$

Why is $\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n$ equal $1/(1-0.7) = 10/3$ ? Can we generalize the above to $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some ...
8k views

### Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
15k views

### Is $dy/dx$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula ...
8k views

### What is 48÷2(9+3)? [duplicate]

Possible Duplicate: Do values attached to integers have implicit parentheses? Implicit multiplication order of operation There is a huge debate on the internet on 48÷2(9+3). I figured if ...
4k views

### Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$

How to prove $$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
1k views

### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can if find the sum for: $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha. I'm more concerned with how to get to to that number. It cites tests to prove ...
4k views

### How can you prove that a function has no closed form integral?

I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction ...
11k views

### Is value of $\pi = 4$?

What is wrong with this? SOURCE
1k views

### How to use the Extended Euclidean Algorithm manually?

I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
1k views

### If $a | m$ and $(a + 1) | m$, prove $a(a + 1) | m$.

Can anyone help me out here? Can't seem to find the right rules of divisibility to show this: If $a | m$ and $(a + 1) | m$, then $a(a + 1) | m$.
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### Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$

For all $a, m, n \in \mathbb{Z}^+$, $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
2k views

### Do odd imaginary numbers exist?

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
1k views

### Is there a name for function with the exponential property $f(x+y)=f(x) \times f(y)$?
I was wondering if there is a name for a function that satisfies the conditions $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \times f(y)$? Thanks and regards!
Sorry for the remedial math but: Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^x$$ For $x = 1$, it's just the classic $n(n+1)/2$. What is it for $x > 1$?