Alternating Series

Part 1 of 7 — The Alternating Series Test

Alternating Series Test (Leibniz Test)

$sum_{n=1}^{infty} (-1)^{n+1} b_n$ converges if:

1. $b_n > 0$ (terms are positive)
2. $b_{n+1} leq b_n$ (decreasing)
3. $lim_{n o infty} b_n = 0$

Examples

sum_{n=1}^{infty} rac{(-1)^{n+1}}{n} = 1 - rac{1}{2} + rac{1}{3} - rac{1}{4} + cdots = ln 2

This is the alternating harmonic series — it converges!

Alternating Series Test 🎯

Key Takeaways — Part 1

AST: decreasing positive terms going to zero → converges.

Alternating Series

Part 2 of 7 — Error Bound

Alternating Series Error Bound

If $S = sum_{n=1}^{infty}(-1)^{n+1}b_n$ and $S_n$ is the $n$th partial sum:

$|S - S_n| leq b_{n+1}$

The error is at most the absolute value of the first omitted term.

Example

S = sum_{n=1}^{infty} rac{(-1)^{n+1}}{n^2}. After 4 terms:

$S_4 = 1 - 1/4 + 1/9 - 1/16$

Error $leq b_5 = 1/25 = 0.04$

Error Bound 🎯

Key Takeaways — Part 2

Error $\leq$ first omitted term. This is a very useful and simple bound!

Alternating Series

Part 3 of 7 — Absolute vs Conditional Convergence

Absolute Convergence

$sum a_n$ converges absolutely if $sum |a_n|$ converges.

Conditional Convergence

$sum a_n$ converges conditionally if $sum a_n$ converges but $sum |a_n|$ diverges.

Key Fact

Absolute convergence → convergence (but NOT vice versa!)

Example

sum rac{(-1)^{n+1}}{n}: converges (AST) but $sum 1/n$ diverges → conditional convergence

sum rac{(-1)^n}{n^2}: $sum 1/n^2$ converges → absolute convergence

Absolute vs Conditional 🎯

Key Takeaways — Part 3

Absolute: $\sum |a_n|$ converges. Conditional: series converges but absolute version doesn't.

Alternating Series

Part 4 of 7 — Approximation with Error Bound

AP Exam Application

"Approximate sum_{n=0}^{infty} rac{(-1)^n}{(2n+1)!} to within $0.001$."

This is $sin(1)$! Compute partial sums until the first omitted term $< 0.001$.

$n=0$: $1$ $n=1$: $1 - 1/6 = 5/6$ $n=2$: $5/6 + 1/120$

Check: $b_3 = 1/5040 approx 0.0002 < 0.001$ ✓

So $S_2 = 5/6 + 1/120 = 101/120 approx 0.8417$ approximates $sin(1)$ within $0.001$.

Error Applications 🎯

Key Takeaways — Part 4

On the AP exam: compute partial sums until the error bound is small enough.

Alternating Series

Part 5 of 7 — Rearrangement Theorem

Riemann Rearrangement Theorem

A conditionally convergent series can be rearranged to converge to ANY number, or to diverge!

An absolutely convergent series gives the same sum regardless of rearrangement.

This is why absolute convergence is "stronger" than conditional convergence.

Rearrangement 🎯

Key Takeaways — Part 5

Conditional → rearrangement changes the sum. Absolute → sum is preserved.

Alternating Series

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Alternating Series — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Alternating Series — Complete! ✅