Advanced High-School Mathematics

(Tina Meador) #1

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


  1. 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.)

  2. 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


  1. 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
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