Guest Posted September 18, 2009 Report Share Posted September 18, 2009 As we all know, the expression [x] refers to the largest integer smaller than or equal to x (also described by the expression "Floor".) What is the integral from zero to infinity of the function exp(-[x])? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 18, 2009 Report Share Posted September 18, 2009 e/(e-1), I don't have a calculator to evaluate that though. Our function is going to look kind of like a set of stairs with steps 1 unit thick. The height of each step being equal to the value of 1/e^k where k is the value of x at the left side of the step. So it's a geometric series: 1/e^0 + 1/e^1 + 1/e^2 + ... + 1/e^n, where n = infinity. So, e*Sum - Sum = e-(1/e^n), n is infinity so 1/e^n is 0 Sum = e/(e-1) Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 18, 2009 Report Share Posted September 18, 2009 e/(e-1), I don't have a calculator to evaluate that though. Our function is going to look kind of like a set of stairs with steps 1 unit thick. The height of each step being equal to the value of 1/e^k where k is the value of x at the left side of the step. So it's a geometric series: 1/e^0 + 1/e^1 + 1/e^2 + ... + 1/e^n, where n = infinity. So, e*Sum - Sum = e-(1/e^n), n is infinity so 1/e^n is 0 Sum = e/(e-1) I agree Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 18, 2009 Report Share Posted September 18, 2009 I fully agree. This one was (believe it) a scholarship question for University Entrance in the 1960's. Is there any fundamental difference, do you think, DeeGee, between integration and summation (apart from the obvious?) Even when one considers the complexities (or simplicities) of contour integration techniques, I would answer in the negative. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 18, 2009 Report Share Posted September 18, 2009 I fully agree. This one was (believe it) a scholarship question for University Entrance in the 1960's. Is there any fundamental difference, do you think, DeeGee, between integration and summation (apart from the obvious?) Even when one considers the complexities (or simplicities) of contour integration techniques, I would answer in the negative. Totally agree (with your negative). Integration is infact summation of tiny parts. A small difference being (Well, I have been out of touch with integration for a long time but I think...)the basic requirement for any integration is continuity of the function. Where as summation doesn't need this condition. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 18, 2009 Report Share Posted September 18, 2009 Oh, you have truly opened up a can of worms with that last comment, DeeGee! The term "piece-wise continuous," however, seems to be used quite frequently. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 21, 2009 Report Share Posted September 21, 2009 1 + e^(-1) + e^(-2) + e^(-3) + ...... = e/(e-1) Quote Link to comment Share on other sites More sharing options...
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As we all know, the expression [x] refers to the largest integer smaller than or equal to x (also described by the expression "Floor".) What is the integral from zero to infinity of the function exp(-[x])?
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