[GSoC] Series: Correcting incorrect limit evaluations based on different assumptions of the limit variable

[sachin-4099]

Fixes: #8481
Fixes: #9041
Fixes: #14556
Fixes: #16722
Fixes: #17792
Fixes: #18992

Brief description of what is fixed or changed

Incorrect limit evaluation takes place based on different assumptions of the variable in the limit expression.

The assumption integer = True is causing these issues.

As we know, Gruntz algorithm changes a variable to a dummy with positive=True if the variable does not possess that property. We can make it define a dummy, if the limit variable has integer=True property.
So, Gruntz algorithm would not need to work with variables having integer=True property and thus this issue won’t occur.
The rewrite line can come later, after the dummy variable has been substituted.

Now the limit evaluations take place correctly:

In [6]: b = Symbol('b', positive=True, integer=True)
In [7]: limit(factorial(b+1)**(1/(b+1)) - factorial(b)**(1/b), b, oo)
Out[7]:  
 -1
e

In [8]: n = Symbol('n',positive=True,integer=True)

In [9]: limit(factorial(n)/sqrt(n)*(E/n)**n,n,oo)
Out[9]:
  ___   ____
\/ 2 *\/ pi


In [10]: z = symbols('z', positive=True)

In [11]: limit(binomial(n + z, n)*n**-z, n, oo)
Out[11]: 1/gamma(z + 1)

In [12]: z = symbols('z', positive=True, integer=True)

In [13]: limit(binomial(n + z, n)*n**-z, n, oo)
Out[13]: 1/gamma(z + 1)

Other Comments

Regression Tests have been added.

Release Notes

• series
  • Adds a condition to limitinf() function of gruntz.py resolving incorrect limit evaluations

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• series
  • Adds a condition to limitinf() function of gruntz.py resolving incorrect limit evaluations (#19292 by @sachin-4099)

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Fixes: #14556
Fixes: #16722 
Fixes: #17792

#### Brief description of what is fixed or changed

**Incorrect limit evaluation** takes place based on **different assumptions** of the variable in the limit expression.

The assumption `integer = True` is causing these issues.

As we know, Gruntz algorithm changes a variable to a dummy with `positive=True` if the variable does not possess that property. We can make it define a dummy, if the limit variable has `integer=True` property. 
So, Gruntz algorithm would not need to work with variables having `integer=True` property and thus this issue won’t occur. 
The rewrite line can come later, after the dummy variable has been substituted.

Now the limit evaluations take place correctly:

```
In [6]: b = Symbol('b', positive=True, integer=True)
In [7]: limit(factorial(b+1)**(1/(b+1)) - factorial(b)**(1/b), b, oo)
Out[7]:  
 -1
e

In [8]: n = Symbol('n',positive=True,integer=True)

In [9]: limit(factorial(n)/sqrt(n)*(E/n)**n,n,oo)
Out[9]:
  ___   ____
\/ 2 *\/ pi


In [10]: z = symbols('z', positive=True)

In [11]: limit(binomial(n + z, n)*n**-z, n, oo)
Out[11]: 1/gamma(z + 1)

In [12]: z = symbols('z', positive=True, integer=True)

In [13]: limit(binomial(n + z, n)*n**-z, n, oo)
Out[13]: 1/gamma(z + 1)
```

#### Other Comments
Regression Tests have been added.


#### Release Notes


<!-- BEGIN RELEASE NOTES -->
* series
  * Adds a condition to `limitinf() function of gruntz.py` resolving incorrect limit evaluations 
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

[sachin-4099]

Please let me know if this is a similar issue: #9252

[codecov]

Codecov Report

Merging #19292 into master will decrease coverage by 0.020%.
The diff coverage is 100.000%.

@@              Coverage Diff              @@
##            master    #19292       +/-   ##
=============================================
- Coverage   75.593%   75.573%   -0.021%     
=============================================
  Files          651       651               
  Lines       169545    169545               
  Branches     40016     40016               
=============================================
- Hits        128165    128131       -34     
- Misses       35762     35795       +33     
- Partials      5618      5619        +1     

[jksuom]

This looks good. Gruntz's algorithm deals with functions of a continuous variable.

[sachin-4099]

Please let me know if this is a similar issue: #9252

What about this? Ping @jksuom

[jksuom]

That looks like a bug. The result should depend on c.

[sachin-4099]

That looks like a bug. The result should depend on c.

Alright.