Quiz: calculating a definite integral

01032014, 09:04 PM
(This post was last modified: 01032014 09:05 PM by Bunuel66.)
Post: #41




RE: Quiz: calculating a definite integral
(01032014 08:39 PM)Thomas Klemm Wrote:(01032014 05:38 PM)Bunuel66 Wrote: This seems to show that having the equality is not enough for keeping it directly after integrating.The problem I see is that \(u=\frac{1}{x}\) is not defined for \(x=0\). The Taylorseries of \(\exp(u)\) is not defined for \(u=\infty\). Don't get the point, \(\exp(\infty)\)=0. The serie is converging whatever the sign of x (more and more slowly as you're closing to 0....). Then we have two expressions who provides similar values whatever the sign of x, and after integration we have a new set of expressions with one which is no more defined on one side (x<0). And as you mention, this is not exactly a Taylor serie in the sense that the sum is not using the derivatives of u(x). The problem is maybe a little bit more subtle (at least for me) than it seems ;(... Regards 

01032014, 09:32 PM
Post: #42




RE: Quiz: calculating a definite integral  
01032014, 11:08 PM
Post: #43




RE: Quiz: calculating a definite integral
Not exactly, but close to. I wrote simple RPN program.
0 STO4 LBL00 RCL3 INPUT +/ Y^X STO+4 RCL1 STO+3 RCL2 RCL3 ?> GT00 RCL4 RCL1 * Stop 0.00005 STO1, 1 STO2 0 STO3 It takes about 2+ min to complete (20000 cycles or 166cycles/sec! ). As HP30b is relatively fast machine  such a 'brute force' method gives acceptable result too. 

01062014, 10:28 AM
Post: #44




RE: Quiz: calculating a definite integral
(01032014 09:04 PM)Bunuel66 Wrote: Don't get the point, \(\exp(\infty)\)=0. The domain of \(\exp(x)\) is \(\mathbb{R}\), but \(\infty \notin \mathbb{R}\). Thus you can not just plug \(\infty\) into the Taylorseries of this function and expect everything works. You can calculate \(\lim_{x\to\infty}\exp(x)\) but that's not the same as \(\exp(\infty)\). This expression is just not defined. HTH Thomas 

01072014, 06:12 PM
Post: #45




RE: Quiz: calculating a definite integral
(01062014 10:28 AM)Thomas Klemm Wrote:(01032014 09:04 PM)Bunuel66 Wrote: Don't get the point, \(\exp(\infty)\)=0. Could have been rewriten as a limit to be more rigorous...;) That said the serie gives the same value than the function also for x<0. Doesn't seems to be the point. And as you mention this is not a Taylor serie strictly speaking. Regards. 

01092014, 07:45 AM
Post: #46




RE: Quiz: calculating a definite integral
(01072014 06:12 PM)Bunuel66 Wrote: Could have been rewriten as a limit to be more rigorous...;) Maybe these posts are helpful:
Cheers Thomas 

01142014, 02:31 PM
Post: #47




RE: Quiz: calculating a definite integral  
01142014, 02:53 PM
Post: #48




RE: Quiz: calculating a definite integral
Nice quiz!
d:) 

01162014, 08:21 PM
Post: #49




RE: Quiz: calculating a definite integral
(12312013 01:14 PM)Thomas Klemm Wrote: It takes 2'27" to calculate this integral on a DM15CC with FIX 9.5 seconds on my outdated iPhone 4s with HP15C emulator by HP: Code: 001 f LBL A 0 ENTER 1 f Integrate > 1.291285997 (blinking, but that's another story) Estimated +4 hours on a real HP15C. As a comparison the following return the same result (no blinking, of course!) in 13.7 and 13.4 seconds, respectively, on my 30year old HP15C: Code:
Cheers, Gerson. 

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