For questions concerning the creation and understanding of pictorial proofs.

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3
votes
0answers
101 views

Using a visual “proof” to show that $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n =1$

The "proof without words" that $\sum_{n=1}^{\infty} \left(\frac 12 \right)^n =1$ is fairly well known: But why can't we apply the exact same logic to $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ ...
30
votes
1answer
540 views

Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In a gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
2
votes
1answer
77 views

On equilateral triangles

I came across the following geometry problem. In the exterior of a triangle $ABC$ three equilateral triangles $ABC' , BCA'$ and $CAB'$ are constructed. Prove that the centroids of these triangles ...
6
votes
2answers
156 views

Explain this calculus proof without words

This demonstrates that $\int_0^1 t^{p/q} + t^{q/p} dt = 1$. Could you please explain how the proof without words shows that?
1
vote
1answer
48 views

I can't figure how to prove this coordinate algebraic geometry proof.

Consider the points $A$ and $C$ are on $y=x^p$. We are told that point $A$ has coordinates $( a, b )$, where $0<a<1$ and point $C$ has coordinates $( c, d )$, where $1<c$. Give a ...
2
votes
0answers
198 views

Looking for proof-without-words of Bezout's identity

I'm looking for a "proof-without-words" of Bezout's identity (for integers). Does anyone know of one?
2
votes
1answer
92 views

Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?

We know that $\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and $\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ ...
4
votes
0answers
512 views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
5
votes
2answers
407 views

Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$ [duplicate]

$$(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$$ I noticed this only because $\displaystyle \sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\displaystyle \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$. But the two things ...
1
vote
2answers
138 views

If $Ax=b$, for $b\ne 0$, has more than one solution, then $Ax=0$ does as well. T or F. Prove this.

I get that this is true, because there's one free variable, so no matter what the augmented matrix is, there always will be an infinite amount of solutions. Right? But how to I explain this as a ...
41
votes
2answers
860 views

Geometric interpretation for sum of fourth powers

Summing the first $n$ first powers of natural numbers: $$\sum_{k=1}^nk=\frac12n(n+1)$$ and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a ...
3
votes
4answers
234 views

Proof-without-words for $\bar a\times (\bar b\times\bar c)=\bar b (\bar a\cdot\bar c)-\bar c (\bar a\cdot \bar b)$ or some visual-biased explanation?

Griffiths' Introduction to Electromagnetism -book has equations called 20.10 below. I have proved this equation d) pretty much on the first mathematics -course I had but I have not yet understood a ...