Infinite Series

Part 1 of 7 — Partial Sums & Geometric Series

Partial Sums

$S_n = sum_{k=1}^n a_k = a_1 + a_2 + cdots + a_n$

$sum_{n=1}^{infty} a_n = L$ means $lim_{n o infty} S_n = L$.

Geometric Series

sum_{n=0}^{infty} ar^n = rac{a}{1-r} quad ext{if } |r| < 1

Diverges if $|r| geq 1$.

Example

sum_{n=0}^{infty} rac{3}{4^n} = rac{3}{1 - 1/4} = 4

Geometric Series 🎯

Key Takeaways — Part 1

Geometric: $\frac{a}{1-r}$ when $|r| < 1$. First ratio, then answer!

Infinite Series

Part 2 of 7 — Telescoping Series & Divergence Test

Telescoping Series

Telescoping & Divergence Test 🎯

Key Takeaways — Part 2

1. Telescoping: most terms cancel
2. Divergence test: $a_n \not\to 0$ → diverges

Infinite Series

Part 3 of 7 — Integral Test & $p$-Series

Integral Test

If $f$ is positive, continuous, decreasing for $x geq 1$, and $a_n = f(n)$:

$sum_{n=1}^{infty} a_n ext{ and } int_1^{infty} f(x),dx ext{ either both converge or both diverge}$

$p$-Series

sum_{n=1}^{infty} rac{1}{n^p} egin{cases} ext{converges} & p > 1 \ ext{diverges} & p leq 1 end{cases}

Integral Test & $p$-Series 🎯

Key Takeaways — Part 3

$p$-series: converges iff $p > 1$. Integral test connects series to improper integrals.

Infinite Series

Part 4 of 7 — Comparison Tests

Direct Comparison Test

For $0 leq a_n leq b_n$:

• $sum b_n$ converges → $sum a_n$ converges
• $sum a_n$ diverges → $sum b_n$ diverges

Limit Comparison Test

If lim_{n o infty} rac{a_n}{b_n} = c where $0 < c < infty$:

$sum a_n$ and $sum b_n$ either both converge or both diverge.

Comparison Tests 🎯

Key Takeaways — Part 4

Direct comparison needs inequality. Limit comparison just needs the ratio limit.

Infinite Series

Part 5 of 7 — Ratio & Root Tests

Ratio Test

Ratio & Root 🎯

Key Takeaways — Part 5

Ratio test for factorials; root test for $n$th powers. Both inconclusive at $L = 1$.

Infinite Series

Part 6 of 7 — Practice Workshop

Mixed Series Practice 🎯

Workshop Complete!

Infinite Series — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Infinite Series — Complete! ✅