anonymous
  • anonymous
Improper integral
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Find the exact value of \[\int\limits_{1}^{\infty} \lfloor x \rfloor e^{-x} dx\]
amistre64
  • amistre64
floor function eh ...
anonymous
  • anonymous
yeah, i am out of idea for the floor function

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anonymous
  • anonymous
maybe some kind of series? just a thought
amistre64
  • amistre64
\[1e^{-x}+2e^{-x}+3e^{-x}+...\]seems like it would find the area
anonymous
  • anonymous
i think that will do it
amistre64
  • amistre64
or integrate each blurb from a to b
amistre64
  • amistre64
|dw:1371137789738:dw|
anonymous
  • anonymous
maybe like this? \[\int\limits_{1}^{2}e^{-x} dx + 2 \int\limits_{2}^{3}e^{-x} dx + 3 \int\limits_{3}^{4} e^{-x} dx + ...\]
anonymous
  • anonymous
i think it is going to look something like \[\frac{e-1}{e^2}+\frac{2(e-1)}{e^3}+\frac{3(e-1)}{e^4}+... \] but not sure exactly how to add this
anonymous
  • anonymous
yeah, that looks good
anonymous
  • anonymous
\[(e-1)\sum_{k=2}^{\infty}\frac{k-1}{e^k}\] or something like that
anonymous
  • anonymous
can we find the value of the series?
anonymous
  • anonymous
wolf gives \(\frac{1}{(e-1)^2}\) for the sum, but not sure how it almost looks geometric maybe it is the derivative of a geometric series?
amistre64
  • amistre64
\[\int\limits_{1}^{2}e^{-x} dx + 2 \int\limits_{2}^{3}e^{-x} dx + 3 \int\limits_{3}^{4} e^{-x} dx + ...\] \[(-e^{-2}+e^{-1}) + 2(-e^{-3}+e^{-2}) + 3(-e^{-4}+e^{-3}) + ...\] this wouldnt be telescoping in some way would it?
anonymous
  • anonymous
oh much better idea!
anonymous
  • anonymous
it's a geometric series!
amistre64
  • amistre64
\[-e^{-2}+e^{-1} -2e^{-3}+2e^{-2}-3e^{-4}+3e^{-3}-4e^{-5}+4e^{-4} ...\]
anonymous
  • anonymous
\[e^{-1} + e^{-2} + e^{-3} + ...\]
amistre64
  • amistre64
:)
anonymous
  • anonymous
so the answer is \[1/(e - 1)\]
anonymous
  • anonymous
yeah that is what i got using long dumb method too
anonymous
  • anonymous
thanks a lot guys, I wish I can give multiple medals
amistre64
  • amistre64
youre welcome
anonymous
  • anonymous
yw

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