Summary: Sum Infinite Series 1 2 Is

Convergence and Divergence of Infinite Series

Basic criteria and precise definitions used in problem solving.

• Infinite series: sum of a sequence, written as Σa_n from n=1 to ∞; study via partial sums S_N = Σ_n=1^N a_n.
• Convergent: S_N → finite limit as N→∞; Divergent: no finite limit.
• nth-term test: if lim_n→∞ a_n ≠ 0, series diverges (necessary condition for convergence).
• Absolute vs conditional convergence: series converges absolutely if Σ|a_n| converges; conditional if original converges but not absolute.

Standard Tests for Convergence

Use these in order of ease/applicability on given series.

• Geometric series: Σ arⁿ converges iff |r| < 1; sum S = a/(1-r).
• p-series: Σ 1/n^p converges ⇔ p > 1 (diverges for p ≤ 1).
• Comparison test: compare with known convergent/divergent series using inequalities.
• Limit comparison: if lim a_n/b_n = L (0<L<∞) then both behave same.
• Ratio test: compute L = lim |a_n+1/a_n|; series converges if L < 1, diverges if L > 1, inconclusive if L = 1.
• Root test: L = lim sup |a_n|^1/n; same conclusions as ratio test; useful for power/exponential terms.
• Alternating series (Leibniz) test: if terms alternate, decrease to 0, series converges; error < next term.
• Cauchy criterion: series converges iff for every ε>0 there exists N with |Σ_k=m+1ⁿ a_k| < ε for all n>m>N.

Summation Techniques and Examples

Practical methods to obtain exact sums or simplify terms.

• Telescoping: use partial fractions to cancel terms, e.g., Σ 1/[n(n+1)] = 1 (sum = 1).
• Partial fractions: standard for rational-term series; decompose to telescoping form.
• Using known expansions: match series to Maclaurin forms of e^x, sin x, cos x, ln(1+x) to get closed sums.
• Abel/Dirichlet methods: rearrangement for series with parameters; use when summing series depending on x.

Power Series and Radius of Convergence

Essential formulas for series in variable x.

• Power series: Σ a_n(x-c)ⁿ; converges for |x-c| < R where R is the radius.
• Radius via root test: R = 1 / lim sup |a_n|^1/n; if limit exists, R = lim |a_n/a_n+1|.
• Taylor/Maclaurin series: expansions of functions around a point; use for evaluating sums at specific x (e.g., alternating harmonic → ln 2).