- Taylor Swift The Teacher
- The function f is defined by the power series f(x) = x - x^3/3 + x^5/5 - x^7/7 + ... + (-1)^n*x^(2n+1)/(2n+1) + ... for all real numbers x for which the series converges.
- Today we will be learning about power series!
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- Using the ratio test, find the interval of convergence of the power series for f. Justify your answer.
- This is part A of the free response question.
- How do we use the ratio test???
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- lim | (-1)^(n+1)*x^(2n+3) * 2n+1 |
- n-inf.
- This is how we use the ratio test. Next we need to find the IOC using | x^2 | 1.
- 2n+3
- How do we find the interval of convergence???
- (-1)^n*x^(2n+1)
- = | x^2 | 1
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- @ x = -1@ x = 1
- Since | x^2 | 1 is -1 x 1, we have to test if f converges at x = -1 and x =1.
- both converge by the alternating series test
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- Show that | f(1/2) - 1/2 | 1/10. Justify your answer.
- This is part B of the free response question.
- How do we show that???
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- Approximating f(1/2) with 1/2 uses the first term of the series, therefore error is... | f(1/2) - 1/2 | (1/2)^3 / 3 = 1/24 1/10using the magnitude of the fist term left off as error bound.
- At x = 1/2, the series for f(x) is an alternating series with terms that decrease in magnitude to 0, therefore a converging alternate series.

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