Expert: Paul Klarreich - 3/9/2009

Question
I'm given the Mclaurin series for sinx (x-x^3/3+x^5/5....) and I have to use
that given series to evaluate the indefinite integral sinx/x as a series. (The first 4
terms of the desired series are sufficient). Thanks for any help.

Answer
Questioner:   Robyn
Category:  Calculus
Private:  No
 
Subject:  Evaluating indefinite integral given the McLaurin series
Question:  I'm given the Mclaurin series for sinx (x-x^3/3+x^5/5....) and I have to use
that given series to evaluate the indefinite integral sinx/x as a series. (The first 4
terms of the desired series are sufficient). Thanks for any help.


sin x =  (x-x^3/3+x^5/5....)

or:
                         x^(2k+1)(-1)^k
sin (x) = SUM[k=0 to inf] --------------
                            (2k+1)!

Divide by x:

sin x                     x^(2k)(-1)^k
------ = SUM[k=0 to inf] --------------
  x                          (2k+1)!
Now integrate each term and get:

                x^(2k+1)(-1)^k
SUM[k=0 to inf] --------------
                (2k+1)(2k+1)!

Now you can write as many terms as you like:  For example, k = 17:
x^35(-1)^35
------------
35(35!)

- x^35
--------
35(35!)

Oh, yes -- you just want the first four.  You can do those.  Start with k = 0.

BTW, it's MACLaurin.  Yes, spelling counts.  It's an automatic 2-point deduction.