Math 243: Calculus II

Unit V Exam: Review Guide

1. Determine whether the sequences converge or diverge. If the sequence converges, find its limit.

a.)		\(\displaystyle a_n = \frac{n^3}{1 + n^2}\)
b.)		\(\displaystyle a_n = \frac{9^{n+1}}{10^n}\)
c.)		\(\displaystyle a_n = \left(1 + \frac{3}{n}\right)^{4n}\)
d.)		\(\displaystyle \left\{ \frac{(-10)^n}{n!} \right\} \)
e.)		\(\displaystyle \left\{ \frac{n\sin n}{n^2 + 1} \right\} \)

2. Determine whether the series converge or diverge. Justify your answer by at least stating which test you use. You should be able to verify the hypotheses of the test, and show that the conclusion holds.

a.)		\(\displaystyle \sum_{n=1}^\infty \frac{n}{n^3 + 1}\)
b.)		\(\displaystyle \sum_{n=1}^\infty \frac{n^2 + 1}{n^3 + 1}\)
c.)		\(\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n+1}}\)
d.)		\(\displaystyle \sum_{n=1}^\infty \frac{1}{\sqrt{n+1}}\)
e.)		\(\displaystyle \sum_{n=1}^\infty \frac{1\cdot 3\cdot 5\cdots (2n - 1)}{5^n\,n!}\)
f.)		\(\displaystyle \sum_{n=1}^\infty \frac{n^{2n}}{(1 + 2n^2)^n}\)
g.)		\(\displaystyle \sum_{n=1}^\infty \frac{\sqrt{n+1} - \sqrt{n-1}}{n}\)

3. Find the sum of the series: \(\displaystyle \ \ \ 1 - e + \frac{e^2}{2!} - \frac{e^3}{3!} +- \cdots \)

4. Express the decimal \( 4.17\overline{326} \) as a fraction by realizing it as the sum of a geometric series.

5. How many terms of the series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^5} \) must be included to ensure that the maximum error of the partial sum is less than \(\dfrac{1}{100000}\)? What is the minimum error of this partial sum?

6. Show that \(\lim \dfrac{n^n}{(2n)!} = 0\) by showing first that the associated series is convergent, and then applying the Test for Divergence.

7. Prove that if the series \(\displaystyle \sum_{n=1}^\infty a_n\) is absolutely convergent, then so is the series \[ \sum_{n=1}^\infty \left(\frac{n+1}{n}\right)a_n.\]

8. Find the MacLaurin series (Taylor series centered at \(x_0 = 0\)) for the functions.

a.)		\(\displaystyle \dfrac{x^2}{1+x} \)
b.)		\(\displaystyle \arctan(x^2) \)
c.)		\(\displaystyle 10^x \)
d.)		\(\displaystyle (1 - 3x)^{-5} \)

9. Use series to evaluate the limit. \[ \lim_{x \to 0} \dfrac{\sin x - x}{x^3} \]

10. The force due to gravity on an object with mass \(m\) at a height \(h\) above the surface of the earth is \[ \mathbf{F} = \dfrac{mgR^2}{(R+h)^2} \] where \(R\) is the radius of the earth and \(g\) is the acceleration due to gravity for an object on the surface of the earth.

a.)		Express \(\mathbf{F}\) as a series in powers of \(\tfrac{h}{R}\).
b.)		Observe that if we approximate \(\mathbf{F}\) by the first term in the series, we get the familiar expression \(\mathbf{F} \approx mg\) that is frequently used when \(h\) is much smaller than \(R\). Use the Remainder Theorem for Alternating Series to estimate the range of values of \(h\) for which the approximation \(\mathbf{F} \approx mg\) is accurate to within one percent. (Use \(R = 6400\) km.)

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