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Why summation and integrate give different results?
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I think I am misunderstanding something. I think that printed results of this code snippet must be equal, but they are not:
import sympy
x = sympy.Symbol('x')
summation = sympy.summation(sympy.exp(-x), (x, 0, sympy.oo))
print(summation) # 1/(-exp(-1) + 1)
integration = sympy.integrate(sympy.exp(-x), (x, 0, sympy.oo))
print(integration) # 1
Can you explain the difference between summation and integrate?
python sympy
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
add a comment |
I think I am misunderstanding something. I think that printed results of this code snippet must be equal, but they are not:
import sympy
x = sympy.Symbol('x')
summation = sympy.summation(sympy.exp(-x), (x, 0, sympy.oo))
print(summation) # 1/(-exp(-1) + 1)
integration = sympy.integrate(sympy.exp(-x), (x, 0, sympy.oo))
print(integration) # 1
Can you explain the difference between summation and integrate?
python sympy
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
add a comment |
I think I am misunderstanding something. I think that printed results of this code snippet must be equal, but they are not:
import sympy
x = sympy.Symbol('x')
summation = sympy.summation(sympy.exp(-x), (x, 0, sympy.oo))
print(summation) # 1/(-exp(-1) + 1)
integration = sympy.integrate(sympy.exp(-x), (x, 0, sympy.oo))
print(integration) # 1
Can you explain the difference between summation and integrate?
python sympy
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
I think I am misunderstanding something. I think that printed results of this code snippet must be equal, but they are not:
import sympy
x = sympy.Symbol('x')
summation = sympy.summation(sympy.exp(-x), (x, 0, sympy.oo))
print(summation) # 1/(-exp(-1) + 1)
integration = sympy.integrate(sympy.exp(-x), (x, 0, sympy.oo))
print(integration) # 1
Can you explain the difference between summation and integrate?
python sympy
python sympy
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
asked Mar 22 at 15:07
SanyashSanyash
2,96821229
2,96821229
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
SymPy's summation function sums the expression with the integers in the given range replacing the variable:
summation(sympy.exp(-x), (x, 0, sympy.oo))
Is equivalent to
exp(0) + exp(-1) + exp(-2) + ...
Which is approximately equal to:
1 + 0.37 + 0.14 + ...
Integration is a different function, which can be interpreted as calculating the area bounded by the curve generated by the function and the x axis (shaded in orange here):
answered Mar 22 at 15:24
ukemiukemi
3,2303728
Oh, I forgot that summation goes over integers only... Thank you.
– Sanyash
Mar 22 at 15:27
add a comment |
Nothing wrong.
If you notice the summation is a sum of geometric sequence, you will find it to be different to integration.
Summation:
sum exp(-x)
= sum exp(-1)^x
= exp(-1)^0 / (1 - exp(-1))
= 1 / (1 - exp(-1))
Integration:
int exp(-x) dx
= - [ exp(-x) ]
= - [ exp(-infinity) - exp(0) ]
= - [ 0 - 1 ]
= 1
Sorry. StackOverflow do not do LaTeX.
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
Hmmm, I don't know what issum of geometric sequence, can you explain this?
– Sanyash
Mar 22 at 15:13
added algebraic derivation
– adrtam
Mar 22 at 15:14
Still can't understand the part where you describe summation derivation... Whysum exp(-1)^xturns toexp(-1)^0 / (1 - exp(-1))?
– Sanyash
Mar 22 at 15:22
en.wikipedia.org/wiki/Geometric_series formula section, the paragraph starts with "As n goes to infinity"
– adrtam
Mar 22 at 15:27
Thank you very much for explanation. Hope you don't worry about accepting another answer, because it points for the root of the problem: I forgot that summation goes over integers only, that's why I thought the results must be equal.
– Sanyash
Mar 22 at 15:31
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
SymPy's summation function sums the expression with the integers in the given range replacing the variable:
summation(sympy.exp(-x), (x, 0, sympy.oo))
Is equivalent to
exp(0) + exp(-1) + exp(-2) + ...
Which is approximately equal to:
1 + 0.37 + 0.14 + ...
Integration is a different function, which can be interpreted as calculating the area bounded by the curve generated by the function and the x axis (shaded in orange here):
answered Mar 22 at 15:24
ukemiukemi
3,2303728
Oh, I forgot that summation goes over integers only... Thank you.
– Sanyash
Mar 22 at 15:27
add a comment |
SymPy's summation function sums the expression with the integers in the given range replacing the variable:
summation(sympy.exp(-x), (x, 0, sympy.oo))
Is equivalent to
exp(0) + exp(-1) + exp(-2) + ...
Which is approximately equal to:
1 + 0.37 + 0.14 + ...
Integration is a different function, which can be interpreted as calculating the area bounded by the curve generated by the function and the x axis (shaded in orange here):
answered Mar 22 at 15:24
ukemiukemi
3,2303728
Oh, I forgot that summation goes over integers only... Thank you.
– Sanyash
Mar 22 at 15:27
add a comment |
SymPy's summation function sums the expression with the integers in the given range replacing the variable:
summation(sympy.exp(-x), (x, 0, sympy.oo))
Is equivalent to
exp(0) + exp(-1) + exp(-2) + ...
Which is approximately equal to:
1 + 0.37 + 0.14 + ...
Integration is a different function, which can be interpreted as calculating the area bounded by the curve generated by the function and the x axis (shaded in orange here):
answered Mar 22 at 15:24
ukemiukemi
3,2303728
SymPy's summation function sums the expression with the integers in the given range replacing the variable:
summation(sympy.exp(-x), (x, 0, sympy.oo))
Is equivalent to
exp(0) + exp(-1) + exp(-2) + ...
Which is approximately equal to:
1 + 0.37 + 0.14 + ...
Integration is a different function, which can be interpreted as calculating the area bounded by the curve generated by the function and the x axis (shaded in orange here):
answered Mar 22 at 15:24
ukemiukemi
3,2303728
edited Mar 22 at 15:27
answered Mar 22 at 15:24
ukemiukemi
3,2303728
answered Mar 22 at 15:24
ukemiukemi
3,2303728
answered Mar 22 at 15:24
ukemiukemi
3,2303728
3,2303728
Oh, I forgot that summation goes over integers only... Thank you.
– Sanyash
Mar 22 at 15:27
add a comment |
Oh, I forgot that summation goes over integers only... Thank you.
– Sanyash
Mar 22 at 15:27
Oh, I forgot that summation goes over integers only... Thank you.
– Sanyash
Mar 22 at 15:27
Oh, I forgot that summation goes over integers only... Thank you.
– Sanyash
Mar 22 at 15:27
add a comment |
Nothing wrong.
If you notice the summation is a sum of geometric sequence, you will find it to be different to integration.
Summation:
sum exp(-x)
= sum exp(-1)^x
= exp(-1)^0 / (1 - exp(-1))
= 1 / (1 - exp(-1))
Integration:
int exp(-x) dx
= - [ exp(-x) ]
= - [ exp(-infinity) - exp(0) ]
= - [ 0 - 1 ]
= 1
Sorry. StackOverflow do not do LaTeX.
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
Hmmm, I don't know what issum of geometric sequence, can you explain this?
– Sanyash
Mar 22 at 15:13
added algebraic derivation
– adrtam
Mar 22 at 15:14
Still can't understand the part where you describe summation derivation... Whysum exp(-1)^xturns toexp(-1)^0 / (1 - exp(-1))?
– Sanyash
Mar 22 at 15:22
en.wikipedia.org/wiki/Geometric_series formula section, the paragraph starts with "As n goes to infinity"
– adrtam
Mar 22 at 15:27
Thank you very much for explanation. Hope you don't worry about accepting another answer, because it points for the root of the problem: I forgot that summation goes over integers only, that's why I thought the results must be equal.
– Sanyash
Mar 22 at 15:31
add a comment |
Nothing wrong.
If you notice the summation is a sum of geometric sequence, you will find it to be different to integration.
Summation:
sum exp(-x)
= sum exp(-1)^x
= exp(-1)^0 / (1 - exp(-1))
= 1 / (1 - exp(-1))
Integration:
int exp(-x) dx
= - [ exp(-x) ]
= - [ exp(-infinity) - exp(0) ]
= - [ 0 - 1 ]
= 1
Sorry. StackOverflow do not do LaTeX.
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
Hmmm, I don't know what issum of geometric sequence, can you explain this?
– Sanyash
Mar 22 at 15:13
added algebraic derivation
– adrtam
Mar 22 at 15:14
Still can't understand the part where you describe summation derivation... Whysum exp(-1)^xturns toexp(-1)^0 / (1 - exp(-1))?
– Sanyash
Mar 22 at 15:22
en.wikipedia.org/wiki/Geometric_series formula section, the paragraph starts with "As n goes to infinity"
– adrtam
Mar 22 at 15:27
Thank you very much for explanation. Hope you don't worry about accepting another answer, because it points for the root of the problem: I forgot that summation goes over integers only, that's why I thought the results must be equal.
– Sanyash
Mar 22 at 15:31
add a comment |
Nothing wrong.
If you notice the summation is a sum of geometric sequence, you will find it to be different to integration.
Summation:
sum exp(-x)
= sum exp(-1)^x
= exp(-1)^0 / (1 - exp(-1))
= 1 / (1 - exp(-1))
Integration:
int exp(-x) dx
= - [ exp(-x) ]
= - [ exp(-infinity) - exp(0) ]
= - [ 0 - 1 ]
= 1
Sorry. StackOverflow do not do LaTeX.
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
Nothing wrong.
If you notice the summation is a sum of geometric sequence, you will find it to be different to integration.
Summation:
sum exp(-x)
= sum exp(-1)^x
= exp(-1)^0 / (1 - exp(-1))
= 1 / (1 - exp(-1))
Integration:
int exp(-x) dx
= - [ exp(-x) ]
= - [ exp(-infinity) - exp(0) ]
= - [ 0 - 1 ]
= 1
Sorry. StackOverflow do not do LaTeX.
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
edited Mar 22 at 15:13
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
answered Mar 22 at 15:11
adrtamadrtam
3,4601321
3,4601321
Hmmm, I don't know what issum of geometric sequence, can you explain this?
– Sanyash
Mar 22 at 15:13
added algebraic derivation
– adrtam
Mar 22 at 15:14
Still can't understand the part where you describe summation derivation... Whysum exp(-1)^xturns toexp(-1)^0 / (1 - exp(-1))?
– Sanyash
Mar 22 at 15:22
en.wikipedia.org/wiki/Geometric_series formula section, the paragraph starts with "As n goes to infinity"
– adrtam
Mar 22 at 15:27
Thank you very much for explanation. Hope you don't worry about accepting another answer, because it points for the root of the problem: I forgot that summation goes over integers only, that's why I thought the results must be equal.
– Sanyash
Mar 22 at 15:31
add a comment |
Hmmm, I don't know what issum of geometric sequence, can you explain this?
– Sanyash
Mar 22 at 15:13
added algebraic derivation
– adrtam
Mar 22 at 15:14
Still can't understand the part where you describe summation derivation... Whysum exp(-1)^xturns toexp(-1)^0 / (1 - exp(-1))?
– Sanyash
Mar 22 at 15:22
en.wikipedia.org/wiki/Geometric_series formula section, the paragraph starts with "As n goes to infinity"
– adrtam
Mar 22 at 15:27
Thank you very much for explanation. Hope you don't worry about accepting another answer, because it points for the root of the problem: I forgot that summation goes over integers only, that's why I thought the results must be equal.
– Sanyash
Mar 22 at 15:31
Hmmm, I don't know what is
sum of geometric sequence, can you explain this?– Sanyash
Mar 22 at 15:13
Hmmm, I don't know what is
sum of geometric sequence, can you explain this?– Sanyash
Mar 22 at 15:13
added algebraic derivation
– adrtam
Mar 22 at 15:14
added algebraic derivation
– adrtam
Mar 22 at 15:14
Still can't understand the part where you describe summation derivation... Why
sum exp(-1)^x turns to exp(-1)^0 / (1 - exp(-1))?– Sanyash
Mar 22 at 15:22
Still can't understand the part where you describe summation derivation... Why
sum exp(-1)^x turns to exp(-1)^0 / (1 - exp(-1))?– Sanyash
Mar 22 at 15:22
en.wikipedia.org/wiki/Geometric_series formula section, the paragraph starts with "As n goes to infinity"
– adrtam
Mar 22 at 15:27
en.wikipedia.org/wiki/Geometric_series formula section, the paragraph starts with "As n goes to infinity"
– adrtam
Mar 22 at 15:27
Thank you very much for explanation. Hope you don't worry about accepting another answer, because it points for the root of the problem: I forgot that summation goes over integers only, that's why I thought the results must be equal.
– Sanyash
Mar 22 at 15:31
Thank you very much for explanation. Hope you don't worry about accepting another answer, because it points for the root of the problem: I forgot that summation goes over integers only, that's why I thought the results must be equal.
– Sanyash
Mar 22 at 15:31
add a comment |
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