Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Question:7. Sometimes the ratio test (Theorem 2) can be applied to a power series containing an infinite number of zero coefficients, provided the zero pattern is regular. Use Theorem 2 to show, for example, that the series
has a radius of convergence , if
and that
has a radius of convergence , if

By making the appropriate assumptions, we can prove that the radius of convergence is for the given series respectively.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Check for the first series

Let the variable ^X2=Z, upon making the substitution the series becomes

It is given that the ratio

In general the radius of convergence is given by | x -x₀|=L .

In this case, the center point^Z_⁰ is 0, so it will be,

Step 2: Check for the second series

Similarly for the series,

We can re-write the equation as

If the series that is formed by taking the x common is convergent then the complete series will also be convergent with the same radius of convergence.

Therefore, the series for which we need to calculate the radius of convergence is

Let the variablex₂= z upon making the substitution the series becomes

It is given that the ratio

In general, the radius of convergence is given by

| x -x₀|=L

In this case, the center point Z₀ is 0, so it will be

By making the appropriate assumptions, we can prove that the radius of convergence is for the given series respectively.

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Question:7. Sometimes the ratio test (Theorem 2) can be applied to a power series containing an infinite number of zero coefficients, provided the zero pattern is regular. Use Theorem 2 to show, for example, that the series
has a radius of convergence , if
and that
has a radius of convergence , if

Step by Step Solution

Step 1:Check for the first series

Step 2: Check for the second series

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Question:7. Sometimes the ratio test (Theorem 2) can be applied to a power series containing an infinite number of zero coefficients, provided the zero pattern is regular. Use Theorem 2 to show, for example, that the serieshas a radius of convergence , ifand thathas a radius of convergence , if

Step by Step Solution

Step 1:Check for the first series

Step 2: Check for the second series

Most popular questions for Math Textbooks

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Question:7. Sometimes the ratio test (Theorem 2) can be applied to a power series containing an infinite number of zero coefficients, provided the zero pattern is regular. Use Theorem 2 to show, for example, that the series
has a radius of convergence , if
and that
has a radius of convergence , if