Limits are plugged in terms become infinite

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Homework Help Overview

The discussion revolves around evaluating the integral from 0 to infinity of the function t^(-1)*e^(-st) dt, focusing on the behavior of the integral as the limits approach infinity and the implications of the series expansion of e^(-st).

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the divergence of the integral, with one noting the series expansion of e^(-st) and questioning the implications of terms becoming infinite when limits are applied. Others explore the behavior of the function near the endpoints and the validity of using asymptotic behavior as proof of divergence.

Discussion Status

The discussion is active, with participants providing insights into the divergence of the integral and suggesting formal approaches such as comparison tests. There is an exploration of different interpretations regarding the proof of divergence, but no consensus has been reached.

Contextual Notes

Participants are considering the behavior of the integral at the lower limit as t approaches 0, particularly in relation to the parameter s and its impact on convergence or divergence.

DylanB
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Homework Statement



find the integral from 0 to infinity of t^(-1)*e^(-st) dt

Homework Equations



e^x series expansion

The Attempt at a Solution



I expanded e^(-st) into its series and then integrated, it appears to diverge since when the limits are plugged in terms become infinite, but I am unsure since the series alternates with (-1)^n.
 
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Sure it diverges. exp(-st) approaches 1 as t->0. So near 0 the integral diverges like 1/t. If s=0 it's exactly 1/t.
 
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Is showing that the function has a diverging asymptote at one of the endpoints strong enough proof to say that it's integral diverges to +inf?
 


Sure. More formally write a comparison test for the integral on the interval of say t from 0 to 1/s. If it's greater than something that diverges, then it diverges.
 

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