1549901369-Elements_of_Real_Analysis__Denlinger_

8.6 Power Series 513

Examples 8.6.18 Maclaurin Series for ln(l + x), tan-^1 x, (

1

) 2 and ( x 2.

l+x l+x)

From our knowledge of geometric series we know that

1

• = ~(-l)kxk = l - x+x^2 - x^3 +x^4 - · · · (21)
  l+x 6
  k=O

everywhere in the interval ( -1, 1). Thus, this series must be the Taylor series

of the function -

1

• about 0- i.e., the Maclaurin series of -
  1
  -. The radius
  l+x l+x
  of convergence of this series is 1, and the series diverges at both endpoints.
  Formula (21) can be used to obtain the Maclaurin series of other functions.
  (a) Recall that when x > -1, ln(l+x ) = j-

1
-dx+C. Integrating the
l+x
series (21) t erm-by-term, we have

J

1 oo (-l)kxk+l

-dx=" +c

l+x (^6) k=O k+l
= ( x - ~2 + ~3 - ~4 + ... ) + c,
which converges for all x in (-1, 1). That is, for all -1<x<1,
ln ( 1 + x ) = x - x2 2 + 3 x3 - x4 4 + · · · + C.
To find the constant C we let x = 0 in this equation, and find that C = 0.
Therefore, for all x in ( -1, 1),
00 ( l)k k+l
ln(l + x) = L -k +xl '
k=O
which must be the Maclaurin series for ln(l + x).
(22)
(b) Recall that tan-^1 x = f-
1

• 2 dx + C. From formula (21) above,
  l+x

_1_ = ~(-x2)k = 1 - x2 + x4 - x6 + x8 -.... (23)

l+x2 6

k=O

By Corollary 8.6.16 and Definition 8.6.17, this must be the Maclaurin series

for -

1

• 
  
  
  
  2. The radius of convergence is again l. Integrating term-by-term,
     l+x
  
  
  
  

00 J ( 00 2k+l)

J

_l _ 2 dx = L (-l)kx2kdx = L(-l)k ;k 1 + C.

l + X k=O k=O +