AP® Calculus BC · Free response

A power series: radius, interval, a derivative and a geometric sum

write it first, then score yourself point by point

Series and Taylor polynomials · Part B: no calculator · 9 points

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The question

Name: ______________________   Date: ____________   A power series: radius, interval, a derivative and a geometric sum (9 points)

Table

The power series for ff

The function ff is defined by the power series f(x)f(x)=∑n=1∞(x−2)nn⋅5n= \sum_{n=1}^{\infty} \frac{(x - 2)^n}{n \cdot 5^n} for all xx where the series converges. Its first three terms are shown.
The first three terms of ∑n=1∞(x−2)nn⋅5n\sum_{n=1}^{\infty} \frac{(x - 2)^n}{n \cdot 5^n}
nn112233
Termx−25\frac{x - 2}{5}(x−2)250\frac{(x - 2)^2}{50}(x−2)3375\frac{(x - 2)^3}{375}

  1. (a) Use the ratio test to find the radius of convergence of the series for ff. [2 points]

  2. (b) Find the interval of convergence of the series for ff. Justify your answer at each endpoint. [3 points]

  3. (c) Find the value of f‴(2)f'''(2). [2 points]

  4. (d) Write the power series for f′(x)f'(x), using sigma notation. Then use it to find the exact value of ∑n=1∞2n−15n\sum_{n=1}^{\infty} \frac{2^{n-1}}{5^n}. [2 points]