Chapter 8: Q. 70 (page 671)

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

Short Answer

Expert verified

The radius of convergence for series $\sum_{k = 0}^{\infty} a_{k} {(c x)}^{k}$ is $\frac{ρ}{|c|}$

Step by step solution

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

Ratio Test for absolute convergence for series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ :

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

As series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ 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 = $ρ$
= $\frac{a_{k}}{a_{k + 1}}$ .

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

Ratio Test for absolute convergence for series $\sum_{k = 0}^{\infty} a_{k} {(c x)}^{k}$ :

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

As series $\sum_{k = 0}^{\infty} a_{k} {(c x)}^{k}$ converges only when,

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

and

therefore

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

Hence Proved.

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Short Answer

Step by step solution

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

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

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Let∑k=0∞akxkbeapowerserieswithapositiveandfiniteradiusofconvergenceρ,andletc≠0.Provethattheseries∑k=0∞ak(cx)khasradiusofconvergenceρc.

Short Answer

Step by step solution

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.