Problems on Convergence of Series and Operations on Series

1. (H) Use the ratio test (Theorem 3 from Section 11.1) to find the radius of convergence of the following power series:

   (a)

   x + 4x² + 9x³ + 16x⁴ + 25x⁵ + ^...

   (b)

   1 + 2x + $${\frac{{4x^{2}}}{{2!}}}$$ + $${\frac{{8x^{3}}}{{3!}}}$$ + $${\frac{{16x^{4}}}{{4!}}}$$ + $${\frac{{32x^{5}}}{{5!}}}$$ + ^...

2. (H) Use the Taylor series listed on page 628 of your textbook and the techniques of adding, subtracting, multiplying, dividing, substituting, differentiating, and integrating to find the first four non-zero terms of the Taylor series centered at 0 for:

   (a)

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

   (b)

   e^xcos x

   (c)

   $${\frac{{1}}{{2}}}$$ln$$\left\vert\vphantom{\frac{x+1}{x-1}}\right.$$$${\frac{{x+1}}{{x-1}}}$$$$\left.\vphantom{\frac{x+1}{x-1}}\right\vert$$

   (Extra Credit)

   arcsin x

3. * Consider the two functions y = e^-x2 and y = 1/(1 + x²).

   (a)

   Write the Taylor expansions for the two functions about x = 0. (You may use the taylor command.) What is similar about the two series? What is different?

   (b)

   Are these functions even or odd, or neither? How might you see this by looking at the series expansions?

   (c)

   By looking at the coefficients, explain why it is reasonable that the series for y = e^-x2 converges for all values of x, but the series for y = 1/(1 + x²) converges only on (- 1, 1). (Note that the ratio test has a problem because some of the terms are zero!)

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Problems on Convergence of Series and Operations on Series

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