Infinite Sequences & Series

🔹 sequences vs series

A sequence {a_n} has a limit L if terms eventually stay arbitrarily close to L (think end behavior).

A series ∑ a_n converges if the sequence of partial sums S_n = a₁ + ⋯ + a_n approaches a finite limit.

🧪 common convergence tests

• divergence: if lim a_n ≠ 0 ⇒ ∑ diverges.
• integral: compare ∑ f(n) to ∫ f(x) dx for positive decreasing f.
• p-series: ∑ 1/n^p converges iff p > 1.
• comparison / limit comparison to a known benchmark.
• alternating: AST if terms decrease to 0 in absolute value → converge; alternating series error bounded by next term.
• ratio: |a_n+1/a_n| → L: L < 1 ⇒ abs converge; L > 1 ⇒ diverge.
• root: ⁿ√|a_n| → L with same L interpretation as ratio.

📚 power series

Use ratio test on absolute values to find open interval for x, then plug endpoints individually.

🌟 Taylor & Maclaurin BC

Maclaurin means a = 0. Know classics: eˣ, sin x, cos x, 1/(1 − x), ln(1 + x).

Error / remainder: Lagrange form uses f⁽ⁿ⁺¹⁾(z)/(n+1)! · (x−a)ⁿ⁺¹ bound on derivatives.

✨ operations

Inside radius: differentiate or integrate series term-by-term. Substitute algebraically to build related series.