# How to plot linear term of a function expanded by Taylor seires in Mathematica?

I am quite new in Mathematica and I am trying to plot a linear term of a particular function in Mathematica. I have tried something like this:

TD = 200
Debye[x_] := 3 (x/TD)^3 NIntegrate[y^3/(Exp[y] - 1), {y, 0, TD/x}]
Fa[x_] := 8.6173324*10^(-5) x (9 *TD/(8 x) + 3 Log[1 - Exp[-TD/x]] -
Debye[x])
Series[Fa[x], {x, 0, 1}]


Actually I want to plot Fa[x] with the linear term expansion. It shows some error called

"NIntegrate::nlim: "y = 200./t is not a valid limit of integration"

However If I just simply plot Fa[x] (without the series syntax), it works. It seems I am missing something regarding the Series syntax. Could anyone please help me out?

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• According to the documentation, you should use Normal[] to make the expression returned by Series[] work. – egwene sedai Feb 1 '16 at 14:50
• Can you please bit specify, I mean where to place that 'Normal[]' command? I tried it but probably I don't know where to write and then how to plot. Could you please write the explicit plot syntax including series command? – baban Feb 1 '16 at 14:52
• I could be wrong but I think Series[] needs a symbolic expression and here you have a numeric one. I think Debye[] has a closed analytic expression, and you can work from there – egwene sedai Feb 1 '16 at 14:56
• I'd recommend two changes. (1) Define Debye as a function that only handles explicitly numeric input. This is done as Debye[x_?NumberQ]:=.... (2) Make ser into a function also restricted to explicit numeric input (and use Normal, as was already noted by @egwenesedai): ser[x_?NumberQ] := Normal[Series[Fa[y], {y, 0, 1}]] /. y -> x. First remember to clear the prior definitions: Clear[Debye,ser]. – Daniel Lichtblau Feb 1 '16 at 15:39

TD = 200;
Debye[x_] :=
3 (x/TD)^3 Integrate[y^3/(Exp[y] - 1), {y, 0, TD/x}, Assumptions -> x > 0]
Fa[x_] := 8.6173324*10^(-5) x (9*TD/(8 x) + 3 Log[1 - Exp[-TD/x]] - Debye[x])
ser = Series[Fa[x], {x, 0, 1}] // Normal

(* 0.00025852 x Log[1 - E^(-200/x)]*)

Plot[ser, {x, 1, 100}] • Wow!! Many thanks @rewi. Your answer just solved what I exactly was looking for. Thank you so much! – baban Feb 1 '16 at 15:51
• you're welcome! – user36273 Feb 1 '16 at 15:52