Infinite Series

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 .

Geometric Series

$sum_{n=0}^{infty} ar^n = \frac{a}{1-r} quad \text{if } |r| < 1$

Diverges if $|r| geq 1$.

Example

$sum_{n=0}^{infty} \frac{3}{4^n} = \frac{3}{1 - 1/4} = 4$

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

sum_{n=1}^{infty}left(\frac{1}{n} - \frac{1}{n+1}\right) = 1

Partial sum: $S_n = 1 - \frac{1}{n+1} \to 1$.

Infinite Series

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

Integral Test

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

Infinite Series

Part 4 of 7 — Comparison Tests

Direct Comparison Test

For $0 leq a_n leq b_n$:

• $sum$ converges → converges

Infinite Series

Part 5 of 7 — Ratio & Root Tests

Ratio Test

L = lim_{n \to infty} left|\frac{a_{n+1}}{a_n}\right|

• $L < 1$: converges absolutely
• $L > 1$ (or $infty$): diverges
• : inconclusive

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! ✅