Convergence and divergence of series — A/L Combined Mathematics

An infinite series ∑_r=1^∞ u_r is investigated by examining the sequence of its partial sums {S_n}, where S_n = ∑_r=1ⁿ u_r. If the limit S = lim_n→∞ S_n exists as a finite real number, the infinite series is said to converge, and S is called the sum to infinity (denoted S_∞). If the limit does not exist or equals ±∞, the series is said to diverge. For an infinite geometric series with first term a and common ratio R, S_∞ = a / (1 - R) if and only if |R| < 1; if |R| ≥ 1, the geometric series diverges.

Check before you answer (A/L Pure Maths Paper 1)

1. Partial Sum Closed Form: Finding the exact algebraic expression for S_n before taking the limit as n → ∞.
2. Condition for Convergence: Stating the strict condition |R| < 1 when computing the sum to infinity of a geometric series.
3. Limit Rigour: Evaluating terms such as lim_n→∞ 1/(n + 1) = 0 or lim_n→∞ Rⁿ = 0 (for |R| < 1) with complete mathematical justification.