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Q-7E

Expert-verifiedFound in: Page 434

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 series**

**has a radius of convergence**** ****, if**

**and that**

**has a radius of ****convergenc****e**** ****, if**

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

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 -x_{0 }|=L .

In this case, the center point^{ }^{Z}_{0} is 0, so it will be,

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^{ }x_{2 }= z upon making the substitution the series becomes

It is given that the ratio

In general, the radius of convergence is given by

| x -x_{0 }|=L

In this case, the center point Z_{0} 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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