Proofs without words collection 1

Posted on 2017-01-03 by Christies Yp
  • \frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots=1;

1fb18070fe9f59184527e4089ef0ebf8 720w

  • \frac{1}{3}=\sum_{i=1}^{\infty}\frac{1}{4^i};

proofwithoutwords1

  • Geometric series: \sum_{n=0}^{\infty}a r^n=\frac{a}{1-r}.

Series

 

  • The sum of odd numbers: \sum_{j=1}^n(2j-1)=n^2.

Xnip2023 03 24 10 02 32

  • An alternating sum of odd numbers: \sum_{j=1}^n(2j-1)(-1)^{n-j}=n.

2020-04-16_21-26

  • Arithmetic Mean-Geometric Mean Inequality: \frac{a+b}{2}\geq \sqrt{ab}.

2020-04-15_11-25

  • Another method:

Screen Shot 2022 10 08 at 15 26 00

  • \sin(x-y)=\sin x\cos y-\cos x\sin y;

Sin

  • Half-angle tangent formula:

$\latex \tan\frac{\theta}{2}=\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}$.

Screen Shot 2022 10 06 at 10 35 45

  • Distance between a point and a line:

Screen Shot 2022 10 06 at 10 49 33

  • Napier’s inequality: If b>a>0, then \frac{a}{b}<\frac{\ln b-\ln a}{b-a}<\frac{1}{a}.

Napier

  • A similar graph to show \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=1, where \ln (1+\frac{1}{n})=\int_1^{1+\frac{1}{n}}\frac{1}{x}dx.

Xnip2023 03 24 10 14 52

  • Jordan’s inequality: when 0\leq x\leq \frac{\pi}{2}, then \frac{2}{\pi}\leq \frac{\sin x}{x}\leq 1.

This can be done by the inequality of length of arc 1,2 and line PQ.

Xnip2023 03 24 10 21 30

  • \left(1+\frac{1}{n}\right)^n<\left(1+\frac{1}{n+1}\right)^{n+1},\forall n\in\mathbb{N}.

Since m_1<m_2<1.

Monotone

 

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