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Chapter 8: Q. 70 (page 671)

$L e t {鈭�}_{k = 0}^{鈭�} a_{k} x^{k} b e a p o w e r s e r i e s w i t h a p o s i t i v e a n d f i n i t e r a d i u s o f c o n v e r g e n c e 蚁, a n d l e t c 鈮� 0 . P r o v e t h a t t h e s e r i e s {鈭�}_{k = 0}^{鈭�} a_{k} {(c x)}^{k} h a s r a d i u s o f c o n v e r g e n c e \frac{蚁}{|c|} .$

Short Answer

Expert verified

The radius of convergence for series ${鈭�}_{k = 0}^{鈭�} a_{k} {(c x)}^{k}$ is $\frac{蚁}{|c|}$

Step by step solution

01

Step. 1 Finding the value of ρ for the series ∑k=0∞akxk

Ratio Test for absolute convergence for series ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$ :

$\lim_{k 鈫� 鈭�} |\frac{a_{k + 1} x^{k + 1}}{a_{K} x^{k}}| = \lim_{k 鈫� 鈭�} |\frac{a_{k + 1} x^{k} x}{a_{K} x^{k}}| = |\frac{a_{k + 1}}{a_{K}} x| . . . . . . . . . . . . (E v a l u a t i n g t h e L i m i t s)$

As series ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$ converges only when, $|x \frac{a_{k + 1}}{a_{K}}| < 1$

and therefore

$|x| < \frac{a_{k}}{a_{k + 1}}$

So, the Radius of convergence = $蚁$
= $\frac{a_{k}}{a_{k + 1}}$ .

02

Step. 2 Finding Radius of Convergence for the Series ∑k=0∞ak(cx)k

Ratio Test for absolute convergence for series ${鈭�}_{k = 0}^{鈭�} a_{k} {(c x)}^{k}$ :

$\lim_{k 鈫� 鈭�} |\frac{a_{k + 1} {(c x)}^{k + 1}}{a_{K} {(c x)}^{k}}| = \lim_{k 鈫� 鈭�} |\frac{a_{k + 1} {(c x)}^{k} (c x)}{a_{K} {(c x)}^{k}}| = |\frac{a_{k + 1}}{a_{K}} c x| . . . . . . . . . . . . (E v a l u a t i n g t h e L i m i t s)$

As series ${鈭�}_{k = 0}^{鈭�} a_{k} {(c x)}^{k}$ converges only when,

$|\frac{a_{k + 1}}{a_{K}} c x| < 1 |x| < \frac{\frac{a_{k + 1}}{a_{K}}}{|c|} |x| < \frac{蚁}{|c|}$

and

therefore

Radius of convergence = $\frac{蚁}{c}$ .

Hence Proved.

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Most popular questions from this chapter

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .

$\sqrt[3]{1 + x}$

If m is a positive integer, how can we find the Maclaurin series for the function $f (c x^{m})$ if we already know the Maclaurin series for the function f(x)? How do you find the interval of convergence for the new series?

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

${鈭�}_{k = 1}^{鈭�} \frac{5}{k} x^{k}$

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .

$\frac{1}{\sqrt{1 + x}}$

What is the definition of an odd function? An even function?

See all solutions

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