Approximation involving an exponential function

[mccoy1]

Homework Statement

I was following a derivation of some laws and I didn't get how they approximate some portion of the expression. That portion/part is exp[gbH/(2kT)]. The book says gbH/2 <<1 and therefore exp[gbH/(2kT)] = 1+gbH/(2kT).

Homework Equations

The Attempt at a Solution

I agree with the value 1, but where did gbH/(2kT) come from? Please help. My understanding is that if gbH/2 is way less than 1, then e.g exp[1.0*10^-15/KT)] = 1.

 

[atomthick]

Because it's the Taylor expansion of exp(x) near 0

exp(x) = 1 + x + (x^2)/2 + (x^3)/6 + ...

You can cut the series at any term you would like, however you can't equal it to 1 because there will be no parameter left to give values to...

 

[HallsofIvy]

You can also get this approximation by replacing the curving graph by a tangent line.

 

[mccoy1]

Because it's the Taylor expansion of exp(x) near 0

exp(x) = 1 + x + (x^2)/2 + (x^3)/6 + ...

You can cut the series at any term you would like, however you can't equal it to 1 because there will be no parameter left to give values to...

Haa, thank you very much. That didn't pop in my head. Thanks a lot.

 

[mccoy1]

You can also get this approximation by replacing the curving graph by a tangent line.

Thanks for that.