Expert: Paul Klarreich - 7/18/2006

Question
I'm in Calc 2, and I have a problem that asks me to "evaluate (Integral sign)e^x/x as an infinite series".

I assume this means use a taylor/maclaurin series to get an estimate for the integral (if that's already wrong, i'm in a pickle).

I figured out that the taylor series for e^x is e^a + e^a(x-a) + e^a(x-a)^2/2! + e^a(x - a)^n/n!
and that is (Sum symbol)e^a(x - a)^n/n!

However i don't know how to find the series for e^x/x, and if I'm supposed to get a number, how I'm to find it.  I'm really quite lost, all your help would be so much appreciated.
-Ashley.

Answer
Hi, Ashley,

Question:  I'm in Calc 2, and I have a problem that asks me to "evaluate (Integral sign)e^x/x as an infinite series".

I assume this means use a taylor/maclaurin series to get an estimate for the integral (if that's already wrong, i'm in a pickle).

I figured out that the taylor series for e^x is e^a + e^a(x-a) + e^a(x-a)^2/2! + e^a(x - a)^n/n!
and that is (Sum symbol)e^a(x - a)^n/n!

However i don't know how to find the series for e^x/x, and if I'm supposed to get a number, how I'm to find it.  I'm really quite lost, all your help would be so much appreciated.
-Ashley.

Suppose you expand the power series about  x = 0, to get the usual series for e^x:

e^x = 1 + x + x^2/2 + ... + x^n/n! + ...

Now divide each term by x:

e^x    1         x          x^(n-1)
--- = --- + 1 + --- + ... + ------- + ...
x     x         2!           n!

Integrate that term by term, to get:
           x^2         x^n
ln x + x + ---- + ... + ---- + ...
          2*2!         n n!

Perhaps this is what you need.