Advanced High-School Mathematics

SECTION 5.4 Polynomial Approximations 295

Further valid sums can be obtained by differentiating.

Exercises

1. Find the Maclaurin series expansion for each of the functions be-
   low:

(a)

1

1 −x^2

(b)

2 x

1 − 2 x^2

(c)

1

(1−x)^2

(d)

1

(1−x^2 )^2

(e) x^2 sinx

(f) sin^2 x (Hint: Use a double-

angle identity.)

(g)

1

(1−x)^3

(h) ln(1 +x^2 )

(i) tan−^14 x

(j) xex

2

2. Find the Maclaurin series expansion for the rational function
   f(x) =
   x+ 1
   x^2 +x+ 1
   . (Don’t try to do this directly; use an appro-
   priate trick.)


3. Sum the following series:

(a)

∑∞

n=0

(x+ 1)n

(b)

∑∞

n=1

n(x+ 1)n

(c)

∑∞

n=0

(−1)nxn

(n+ 1)!

(d)

∑∞

n=1

(−1)n+1xn

n

(e)

∑∞

n=1

(−1)n+1x^2 n

n

(f)

∑∞

n=1

n^2 xn

4. Sum the following numerical series:

(a)

∑∞

n=0

(−1)nπ^2 n+1

(2n+ 1)!

(b)

∑∞

n=1

(−1)n+1(e−1)n

n

(c)

∑∞

n=0

(−1)n(ln 2)n

(n+ 1)!

(d)

∑∞

n=1

(−1)n+1

n 22 n

(e)

∑∞

n=1

n^2

3 n