AP® Calculus BC · Free response

Four series: a geometric sum, the nth term test, the integral test and a p-series

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

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The question

Name: ______________________   Date: ____________   Four series: a geometric sum, the nth term test, the integral test and a p-series (9 points)

Table

Four series

The four series below are used in parts (a) through (d). No calculator is allowed.

The series for each part (kk is a positive constant)
PartSeries
(a)∑n=2∞3(−25)n\sum_{n=2}^{\infty} 3\left(-\frac{2}{5}\right)^n
(b)∑n=1∞n23n2+1\sum_{n=1}^{\infty} \frac{n^2}{3n^2 + 1}
(c)∑n=1∞ne−n2\sum_{n=1}^{\infty} ne^{-n^2}
(d)∑n=1∞nnk\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^{k}}

  1. (a) The series in part (a) is geometric. Explain why it converges, and find its sum. [2 points]

  2. (b) Show that the series in part (b) diverges. [2 points]

  3. (c) Use the integral test to determine whether the series in part (c) converges or diverges. Show that the conditions for the test are met. [3 points]

  4. (d) Find all values of kk for which the series in part (d) converges. Justify your answer. [2 points]