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Q-7E

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Found in: Page 434

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# 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

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

See the step by step solution

## 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 Z0 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 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 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.