f(x)=1/(1+x)^2/3 (a) find the first 5 terms of the taylor series centered at 0.

Question

f(x)=1/(1+x)^2/3 (a) find the first 5 terms of the taylor series centered at 0. (b) find the first 5 terms of the taylor series centered at 7. (c) use both polynomials to estimate (4.5)^-2/3. use at least 5 decimals. (d) Calculate absolute error for both answers. Use at least 5 decimals. Which approximation was better? Is this what you would expect (e) graph f(x), f'(x), and f''(x). Be sure that both 0 and 7 are included in the frame. Does this help explain why one approximation was better? how?

Explanation / Answer

The Taylor series about 0 for sin x is sin (x) = ?(-1)^n * x^(2n + 1) / (2n + 1)! So, to get the Taylor series about 0 for sin (x^2), plug in x^2 to both sides of the above: sin (x^2) = ?(-1)^n (x^2)^(2n + 1) / (2n + 1)! Simplifying gives sin (x^2) = ?(-1)^n * x^(4n + 2) / (2n + 1)! Now, here's what they mean by "nonzero" terms. A Taylor series (or general power series) is usually written in the form c_0 + c_1x + c_2x^2 + c_3x^3 + .... In your series, if you plug in n = 0, n = 1, and n = 2, you get something times x^2, something times x^6, and something times x^10. So you can think of the series as 0 + 0x + c_2x^2 + 0x^3 + 0x^4 + 0x^5 + c_6x^6 + 0x^7 + 0x^8 + 0x^9 + c_10x^10 + ... So the coefficients of x^2, x^6, and x^10 will be the first three nonzero terms. In part (d), just integrate the sum of those three terms and you'll have the estimate you want.

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