22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!
2 Sum of a geometric series ★★★
Find the sum of \(3+1.5+0.75+0.375+\cdots\).
3 Use the divergence test ★★★
Show that \(\sum_{n=1}^{\infty}\dfrac{n}{n+4}\) diverges.
4 Classify three p-series ★★★
Decide whether each series converges: (a) \(\sum \dfrac1{n^3}\); (b) \(\sum \dfrac{1}{\sqrt n}\); (c) \(\sum n^{-2/3}\).
5 A repeating decimal ★★★
Write \(0.272727\ldots\) as a geometric series and find the fraction it equals.
6 A telescoping partial sum ★★★
Let \(s_N\) be the partial sums of \(\sum_{k=1}^{\infty}\dfrac{1}{k(k+1)}\). Use \(\dfrac1{k(k+1)}=\dfrac1k-\dfrac1{k+1}\) to find \(s_4\) and the sum of the series.
7 Radius of a simple power series ★★★
Find the radius of convergence of \(\sum_{n=0}^{\infty}\dfrac{x^n}{2^n}\) and its sum at \(x=1\).
8 A bouncing ball ★★★
A ball is dropped from 10 feet (about 3.05 m). After each bounce it rises to 60% of its previous height. What total distance does it travel before coming to rest?
9 Ratio test with powers ★★★
Use the ratio test on \(\sum_{n=1}^{\infty}\dfrac{n^2}{3^n}\).
10 Two comparisons ★★★
Decide whether each series converges: (a) \(\sum \dfrac{1}{n^2+5}\); (b) \(\sum \dfrac{1}{2n-1}\).
11 Integral test with an exponential ★★★
Show that \(\sum_{n=1}^{\infty} n e^{-n^2}\) converges by the integral test, and evaluate the integral you used.
12 How many terms are enough? ★★★
The series \(\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n^2}\) converges. How many terms guarantee an error below 0.1? Give the resulting estimate \(s_N\).
13 Absolute or conditional? ★★★
Classify \(\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{\sqrt n}\) as absolutely convergent, conditionally convergent or divergent.
14 A Maclaurin approximation ★★★
Write the first four terms of the Maclaurin series of \(e^{2x}\) and use them at \(x=0.1\) to approximate \(e^{0.2}\).
15 True or false? ★★★
Decide whether each statement is true or false and justify. (a) If \(a_n\to 0\), then \(\sum a_n\) converges. (b) If \(\sum a_n\) converges, then \(a_n\to 0\). (c) If \(\sum |a_n|\) converges, then \(\sum a_n\) converges. (d) \(\sum (-1)^n\) converges because it alternates.
16 Interval of convergence ★★★
Find the interval of convergence of \(\sum_{n=1}^{\infty}\dfrac{x^n}{n\,3^n}\), testing both endpoints.
17 A telescoping series with partial fractions ★★★
Find \(\sum_{n=1}^{\infty}\dfrac{2}{n(n+2)}\).
18 When the ratio test fails ★★★
Show that the ratio test gives \(L=1\) for both \(\sum\dfrac1n\) and \(\sum\dfrac1{n^2}\), then decide each series another way. What does this prove about the case \(L=1\)?
19 Root test ★★★
Does \(\sum_{n=1}^{\infty}\left(\dfrac{2n+1}{3n+4}\right)^n\) converge?
20 Estimating a logarithm ★★★
Use the series \(\ln(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots\) to approximate \(\ln 1.1\) with three terms, and bound the error.
21 Differentiating a power series ★★★
Starting from \(\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^n\) for \(|x| < 1\), show that \(\sum_{n=1}^{\infty} n x^n=\dfrac{x}{(1-x)^2}\) and evaluate \(\sum_{n=1}^{\infty}\dfrac{n}{2^n}\).
22 Factorial and power ★★★
Use the ratio test on \(\sum_{n=1}^{\infty}\dfrac{n!}{n^n}\).
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