<p>I need help with a problem in Peterson's AP Calculus. In the 3rd test, problem 29, how did they know that the series given was the Taylor series for 1/X?</p>
<p>The series was: (-1)^(n)*(x-1)^(n).</p>
<p>That series is geometric. If you expand it out, or by using rules of exponents, you can see that the common ratio is -(x - 1) or just (1 - x). Because a geometric series can be expressed by a/(1-r), where a is the first term and r the common ratio between terms, you can substitute (1-x) for r and get a/(1 - (1 - x)) which simplifies to a / x. I'm assuming the fist term of the series was 1 (n = 0), which means it just is 1/x.</p>
<p>Thanks. I have another question. In the Kaplan AP Calculus book's last practice test, the value for C in part D of question 6 changes from 1/3 to 3. Is this correct? Also, how do they get from the series X^n/3^(n-1) to X^(n+1)/3^n?</p>