Series with Arbitrary Signs Flashcards (29 cards)

How are the terms of a real alternating series expressed using a positive sequence?

$u_n=(-1)^n a_n$ or $u_n=(-1)^{n+1}a_n$, where $a_n=|u_n|>0$.

What conditions ensure convergence under the Alternating Series Test for $u_n=(-1)^n a_n$?

$a_n\geq 0$ is eventually decreasing and tends to zero.

When does the alternating Riemann series converge?

It converges if and only if $\alpha>0$.

When is a numerical series $\sum u_n$ absolutely convergent?

When $\sum |u_n|$ converges.

What does convergence of $\sum |u_n|$ imply about $\sum u_n$?

$\sum u_n$ converges.

What bound holds when $\sum |u_n|$ converges?

$\left|\sum_{n=0}^{\infty}u_n\right|\leq\sum_{n=0}^{\infty}|u_n|$.

What does conditional convergence mean for a series and its absolute-value series?

The series converges, but its series of absolute values diverges.

What should be checked first when determining the nature of an arbitrary-sign series?

Check whether $u_n\to0$.

After confirming $u_n\to0$, what series should be tested?

Test the absolute-value series $\sum|u_n|$.

If the absolute-value series diverges, what tests or methods follow?

Check the Alternating Series Test; otherwise use an asymptotic expansion of the general term.

How do consecutive term signs behave in an alternating series?

Consecutive terms have opposite signs.

How is the remainder magnitude bounded for a convergent alternating series?

$|R_n|\leq a_{n+1}=|u_{n+1}|$.

What determines the sign of the remainder in a convergent alternating series?

It has the same sign as the first omitted term, $u_{n+1}$.

Between which partial sums does the sum of a convergent alternating series lie?

$S_{2n+1}\leq S\leq S_{2n}$.

Why does $\sum_{n}\frac{(-1)^n}{n+1}$ converge by the Alternating Series Test?

Its magnitudes $\frac{1}{n+1}$ decrease to zero.

When does the standard Riemann series converge?

It converges if and only if $\alpha>1$.

Why does the alternating Riemann series diverge when $\alpha\leq0$?

Its terms do not tend to zero.

What series should be tested for absolute convergence with arbitrary signs?

The positive-term series $\sum |u_n|$.

Which tests can establish convergence of $\sum |u_n|$?

Positive-series tests such as the Riemann, comparison, ratio, or root test.

Why does $\sum_{n\geq1}\frac{\cos n}{n^2}$ converge absolutely?

Its absolute terms are bounded by $1/n^2$, whose series converges.

What does the absolute-term ratio tend to for $u_n=\frac{(-1)^n n^3}{n!}$?

Zero.

Why does $\sum_{n\geq0}\frac{(-1)^n n^3}{n!}$ converge absolutely?

Its absolute-value series converges by the Ratio Test.

Why is the alternating harmonic series conditionally convergent?

It converges, while the harmonic series of absolute values diverges.

Which test establishes convergence of the alternating harmonic series?

The Alternating Series Test.

Why is $\sum_{n\geq1}(-1)^n/\sqrt n$ conditionally convergent?

It converges, while $\sum_{n\geq1}1/\sqrt n$ diverges.

Which test establishes convergence of $\sum_{n\geq1}(-1)^n/\sqrt n$?

The Alternating Series Test.

For $u_n=(-1)^n(\sqrt{n+1}-\sqrt n)$, what does rationalization give for $|u_n|$?

$|u_n|=\frac{1}{\sqrt{n+1}+\sqrt n}\sim\frac{1}{2\sqrt n}$.

For $u_n=(-1)^n(\sqrt{n+1}-\sqrt n)$, what is the convergence status of its absolute-value series?

It diverges.

For $u_n=(-1)^n(\sqrt{n+1}-\sqrt n)$, what is the convergence status of the series?

It is conditionally convergent.