The goal of this notebok is to check the veracity of the following important information you may have seen on social networks



We will reproduce the previous equation, step by step as:
$$
\frac{\pi e^{\frac{A}{B}} - e^{i E}}{C D}
$$


```python
from sympy import *
init_printing(use_latex=True) 
t = Symbol('t')
k = Symbol('k')
n = Symbol('n')
```


```python
A=Limit(2**(2*n) * factorial(n)**2 * log(7) / factorial(2*n) / sqrt(n), n, +oo)
A
```


```python
A.doit()
```


```python
B=Integral(exp(-t**2), (t,0,+oo))
B
```


```python
B.doit()
```


```python
C = Integral(3 / (t**6 + 1), (t,0,+oo))
C

```


```python
C.doit()
```


```python
D = Integral(exp(-pi*t**2), (t,-oo,+oo))
D
```


```python
D.doit()

```


```python
E=Sum( 8*pi/(4*k+1)/(4*k+3), (k,0,+oo))
E

```


```python
E.doit().simplify()
```

# Conclusion


```python
total=( pi*exp(A.doit()/B.doit()) - exp(I*(E).doit().simplify()) )/ C.doit() / D.doit()
total.simplify()
```

Which leads me to the conclusion the expression is false ! In fact, it seems that the sum expression shoud not have a $\pi$ in it, and that the both exponentials should be factorized by $\pi$. 

The original web page with the mistake is http://www.brouty.fr/Maths/anniv.html.


```python
total2=( pi*(exp(A.doit()/B.doit()) - exp(I*(E/pi).doit().simplify()) )) / C.doit() / D.doit()
total2
```

2c15e922c552679229ba09c7b7202dad1cddebdf
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Jupyter Notebook
notebooks/Checking Equations With SimPy.ipynb
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MIT
notebooks/Checking Equations With SimPy.ipynb
Hash--/documents
86a2ba249a3a478cba9bdcd511d02c4f4302d6fc
MIT
notebooks/Checking Equations With SimPy.ipynb
Hash--/documents
86a2ba249a3a478cba9bdcd511d02c4f4302d6fc
MIT
Qwen/Qwen-72B
1. YES
2. YES