Series help!!! please!

[bobbob_924]

9. Power series for f (x) is given by 1? x + x2 ? x3 +...+ (?1)^k x^k

a. Find the radius of convergence and the sum of the power series f (x) . Express your answer as
in ? form.

b. Substitute x with x^2 in the expression for the sum you found in part a. Find the radius of
convergence of this new power series.

c. Integrate both sides of the new power series you found in part b. Define the Taylor series
showing the first three nonzero terms and the general term.

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10. The function f has derivative of all orders for all real numbers x. (9 points)
Assume (3) 5, f = f '(3) = ?3, f ''(3) =1,and f '''(3) = ?4.
a. Write the third-degree Taylor polynomial for f about x=3 and used it to approximate f (3.3) .
b. The fourth derivative of f satisfies the inequality f 4 (x) ? 5 for all x in the closed interval
[3.3,3.5] . Use the Lagrange error bound on the approximation to f (3.3) found in part a to
explain why f (3.3) ? 4.5 .
c. Write the fourth-degree Taylor polynomial, P(x), for g(x) = f (x2 + 3) about x=0. Explain if g has
relative max or min at x=0.

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8. The Maclaurin series for f (x) is given by
2+(x)2^2/2!+(x^2)2^3/3!....+(x^k-1)2^k/k!
a. Find f '(0), f "(0) and f^7 (0)
b. For what values of x does the given series converge? Explain your reasoning.
c. Let g(x) = xf (x) . Write the Maclaurin series for g(x), showing the first three nonzero
terms and the general term.
d. Write g(x) in terms of a familiar function without using series. Then, write f (x) in
terms of the same familiar function.

 

[galactus]

9. Power series for f (x) is given by 1? x + x^2 ? x^3 +...+ (?1)^k x^k

a. Find the radius of convergence and the sum of the power series f (x) . Express your answer as
in ? form.

This is the closed form for the alternating geometric series. Remember the closed form for \(\displaystyle \sum_{k=0}^{\infty}x^{k}=\frac{1}{1-x}\)

Since this is alternating, \(\displaystyle \sum_{k=0}^{\infty}(-1)^{k}x^{k}=\frac{1}{1-(-x)}=\frac{1}{1+x}\)

Use the ratio test to find the interval of convergence. The series is centered at x=0.

\(\displaystyle \frac{(-1)^{k+1}x^{k+1}}{(-1)^{k}x^{k}}=|x|\)

The series converges if \(\displaystyle -1<x<1\).

Thus, the radius of convergence is R=1.

b. Substitute x with x^2 in the expression for the sum you found in part a. Find the radius of
convergence of this new power series.

Give it a go. What is the closed form for \(\displaystyle \sum_{k=0}^{\infty}(-1)^{k}x^{2k}\)?.

c. Integrate both sides of the new power series you found in part b. Define the Taylor series
showing the first three nonzero terms and the general term.

Upon integrating the above, we get \(\displaystyle \sum_{k=0}^{\infty}\frac{(-1)^{k}x^{2k+1}}{2k+1}\)

This is the Taylor series for what function?.