– What does it mean for a series to converge? How can you check this? - What does it mean for a series to converge uniformly? What do you have to show to prove it does not converge uniformly? - Show that if \(\sum_{n\in \NN} a_n < \infty\) converges, then \[a_n \ctz{n}\]. - Show that convergent sequences have small tails in the following sense: \[\sum_{n > N} a_n \ctz{N}\]. - Is this a necessary and sufficient condition for convergence? - State the ratio, root, integral, and alternating series tests. - Prove that the harmonic series diverges - Derive a formula for the sum of a geometric series. - State and prove the \(p\dash\)test. - What does it mean for a series to converge absolutely? - Find a sequence that converges but not absolutely.
Solution
1. What convergence of a series means, and how to check it.
\(\sum_{n\ge 1}a_n\) converges iff the sequence of partial sums \(s_N = \sum_{n\le N}a_n\) converges (to a finite limit). Practical checks: (i) necessary: \(a_n \to 0\); (ii) Cauchy criterion: for every \(\eps > 0\) there is \(N\) with \(|\sum_{n=M+1}^{K}a_n| < \eps\) for all \(K > M \ge N\); (iii) tests: comparison, ratio, root, integral, alternating series, and absolute convergence.
2. Uniform convergence of a series, and how to disprove it.
\(\sum f_n\) converges uniformly on \(E\) iff the partial sums \(S_N = \sum_{n\le N}f_n\) converge uniformly on \(E\): \(\sup_{x\in E}|S_N(x) - S(x)| \to 0\). To show it does NOT converge uniformly: exhibit \(\eps > 0\) and points \(x_N \in E\) with \(|S_N(x_N) - S(x_N)| \ge \eps\) for infinitely many \(N\) (or: the tail \(\sup_{x\in E}|\sum_{n>N}f_n(x)|\) fails to tend to \(0\)).
3. If \(\sum_n a_n\) converges, then \(a_n \to 0\).
\(a_n = s_n - s_{n-1} \to s - s = 0\).
4. Convergent sequences have small tails; this is necessary and sufficient.
\(\sum_{n\ge1}a_n\) converges iff the tails \(T_N = \sum_{n>N}a_n\) satisfy \(T_N \to 0\) as \(N \to \infty\) (this is just \(s_N \to s\), with \(T_N = s - s_N\)). So yes, it is a necessary and sufficient condition for convergence (of the partial sums themselves); note this concerns series, and it is equivalent to the Cauchy criterion.
5. Ratio test.
if \(a_n \ne 0\) eventually and \(\limsup |a_{n+1}/a_n| < 1\), then \(\sum a_n\) converges absolutely; if \(\liminf |a_{n+1}/a_n| > 1\), it diverges. Proof of the first: pick \(r\) with \(\limsup < r < 1\); then \(|a_n| \le C r^n\) eventually, and the geometric series \(\sum r^n\) converges.
6. Root test.
if \(\limsup \sqrt[n]{|a_n|} < 1\), then \(\sum a_n\) converges absolutely (choose \(r\) with the limsup \(< r < 1\); eventually \(\sqrt[n]{|a_n|} \le r\), i.e. \(|a_n| \le r^n\)); if \(\limsup\sqrt[n]{|a_n|} > 1\), the terms do not tend to \(0\) and the series diverges.
7. Integral test.
if \(f \ge 0\) is decreasing on \([1,\infty)\), then \(\sum_{n=1}^N f(n)\) and \(\int_1^N f\) converge or diverge together, since \(\int_1^{N+1}f \le \sum_{n=1}^N f(n) \le f(1) + \int_1^N f\) (compare the sum to the integral over unit intervals).
8. Alternating series test.
if \(a_n \ge 0\) decreases to \(0\), then \(\sum (-1)^{n+1}a_n\) converges: the even and odd partial sums are monotone and bounded (interleaved), so both converge to a common limit, and the error after \(N\) terms is bounded by \(a_{N+1}\).
9. The harmonic series diverges.
group terms: \(1 + \tfrac12 + (\tfrac13+\tfrac14) + (\tfrac15+\cdots+\tfrac18) + \cdots \ge 1 + \tfrac12 + \tfrac12 + \tfrac12 + \cdots = \infty\); or use the integral test with \(f(x) = 1/x\) (\(\int_1^\infty dx/x = \infty\)).
10. Sum of a geometric series.
\(s_N = \sum_{n=0}^{N-1} x^n\) satisfies \((1-x)s_N = 1 - x^N\), so \(s_N = \frac{1-x^N}{1-x}\) for \(x \ne 1\); if \(|x| < 1\), \(s_N \to \frac{1}{1-x}\). If \(|x| \ge 1\) and \(x \ne 1\), the terms do not tend to \(0\), so it diverges; at \(x = 1\), \(s_N = N \to \infty\).
11. The \(p\)-test.
\(\sum_{n\ge1} n^{-p}\) converges iff \(p > 1\). For \(p > 1\) use the integral test: \(\int_1^\infty x^{-p}dx = \frac{1}{p-1} < \infty\); for \(p \le 1\): \(\sum n^{-p} \ge \sum n^{-1} = \infty\) (or integral test for \(p < 1\)).
12. Absolute vs. conditional convergence.
\(\sum a_n\) converges absolutely iff \(\sum |a_n| < \infty\), which implies convergence (by the Cauchy criterion: \(|\sum_{M+1}^K a_n| \le \sum_{M+1}^K |a_n|\)). Conditional convergence means \(\sum a_n\) converges but \(\sum|a_n| = \infty\). Example: \(\sum (-1)^{n+1}/n\) converges (alternating series test, 8) but \(\sum 1/n\) diverges (<1>9).
13. Q.E.D.