VA HW 19: Power Series, Part X C 1)Find A Power Series Rep X C https:// /web/Stu

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VA HW 19: Power Series, Part X C 1)Find A Power Series Rep X C https:// /web/Student/Assignment-Responses/submit?dep 13837341 www.webassign.net. Apps D Physics D Math 1080 (online), su & MATH 1080 Course Pa D Patrick MT wolframlAlpha Widge Find a power series representation for f. 1 x x) In 2x f(x) Graph f and several partial sums sn(x) on the same screen What happens as n increases? (Enter your answer using interval notation.) As n increases sn(x) approximates f better on the interval of convergence which is l HE Search the web and Windows 3:14 PM 7/25/2016

Explanation / Answer

Use the rule ln A/B = ln A - ln B to write

ln (1 + x)/(1 - x) = ln (1 + x) - ln (1 - x).

Recall that for |x| < 1, ln (1 + x) = (n = 1, ) (-1)^(n - 1) x^n/n. Then

ln (1 - x) = (n = 1, ) (-1)^(n - 1) (-x)^n/n

= (n = 1, ) (-1)^(n - 1) (-1)^n x^n/n = -(n = 1, ) x^n/n.

So

ln (1 + x) - ln (1 - x) = (n = 1, ) (-1)^(n - 1) x^n/n + (n = 1, ) x^n/n

= (n = 1, ) [(-1)^(n - 1) + 1] x^n/n.

When n is even, n - 1 is odd, so (-1)^(n - 1) + 1 = (-1) + 1 = 0. When n is odd, n - 1 is even, so (-1)^(n - 1) + 1 = 1 + 1 = 2. Therefore

(n = 1, ) [(-1)^(n - 1) + 1] x^n/n = (n = 1, , n odd) 2 x^n/n

= (n = 1, ) 2x^(2n-1)/(2n-1)

The power series representation for ln (1 + x)/(1 - x) (for |x| < 1) can now be written

ln (1 + x)/(1 - x) = (n = 1, ) 2x^(2n-1)/(2n-1) = 2x + 2x^3/3 + 2x^5/5 + …

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