Convergence Tests for Series, Function Series, and Infinite Products

All flashcards

What does the nth-term test show when $a_n$ does not approach zero?

The series $\sum a_n$ diverges; if $a_n\to0$, the test is inconclusive.

How does the ratio test use liminf and limsup?

For $\ell=\liminf|a_{n+1}/a_n|$ and $L=\limsup|a_{n+1}/a_n|$, $L<1$ gives absolute convergence and $\ell>1$ gives divergence.

What quantity does the root test examine?

It examines $r=\limsup_{n\to\infty}\sqrt[n]{|a_n|}$; $r<1$ gives absolute convergence and $r>1$ gives divergence.

When does an alternating series converge by the alternating series test?

When its positive term magnitudes are nonincreasing and approach zero.

What distinguishes conditional convergence from absolute convergence?

A conditionally convergent series converges, but its series of absolute values diverges.

Quiz questions

1. Suppose $|a_n|\leq b_n$ for all sufficiently large $n$, where $b_n\geq0$, and $\sum b_n$ converges. What follows?Answer: The series $\sum a_n$ converges absolutely.
2. For positive terms $a_n$ and $b_n$, suppose $\lim_{n\to\infty}a_n/b_n=c$ with $0<c<\infty$. What does limit comparison establish?Answer: The two series either both converge or both diverge.
3. For which value of $p$ does the p-series $\sum_{n=1}^{\infty}1/n^p$ converge?Answer: $p=\frac{3}{2}$
4. Which conditions are sufficient for convergence by the alternating series test?Answer: The term magnitudes are nonincreasing and tend to zero.
5. Suppose $|f_n(x)|\leq M_n$ for every $x$ in a set $A$, and $\sum M_n$ converges. What does the Weierstrass M-test imply?Answer: $\sum f_n(x)$ converges absolutely and uniformly on $A$.