1549901369-Elements_of_Real_Analysis__Denlinger_

8. 7 Analytic Functions 533

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To see that 2:: Ri and 2:: Cj converge absolutely, observe that 'Vn E .N,

i=l j=l

i~ IRil = i~ lj~i aijl ~ i~ c~ 1 laij1) ~ i~ C~ 1 laij1) and

Since their partial sums are bounded, these nonnegative series converge. •

EXERCISE SET 8. 7

l. Complete the proof of Theorem 8 .7 .3 by deriving (c) from (b).

2. Consider the polynomial function p(x) = 3x^4 - 5x^3 + x^2 - 8.

(a) Prove that pis analytic at O; find its Maclaurin series and the radius

of convergence.

(b) Prove that p is analytic at 2; find its Taylor series about 2 and the

radius of convergence. Simplify the result to show that it equals p( x).

(c) Is p analytic everywhere? Justify your answer.

3. On the basis of Exercise 2, state a theorem about polynomials, their
   analyticity, their Maclaurin series, and their Taylor series about c =J. 0.


4. Verify that the Maclaurin polynomials for sin x are as given in Example
   8.7.4 (b).


5. Prove the claim made in Example 8.7.4 (c).


6. Use the results of Example 8.7.4 and the algebra of power series (8.6.9) to
   find Maclaurin series for the "hyperbolic" functions, sinh x = ~ (ex -e-x)
   and coshx = ~(ex+ e- x). Find their intervals of convergence and prove
   that these series converge to these functions everywhere in these intervals.
   Compare these series with the series for sin x and cos x.


7. Use known power series and the methods of this section to derive Maclau-
   rin series representations for each of the following functions. In each case
   find the interval of convergence.
   (a) x^2 ex
   ( c) sin x + cos x
   (e) cos^2 x [Use trig. identity.]

(b) x^3 sinx

(d) xln(l +x)

(f) sin^2 x