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Theorems 2 and 3: ε, δ Condition for Limits
Like other types of limits, limits of sequences have an ε, δ condition. It will be used later to prove theorems on series. THEOREM 2 (ε, δ Condition for Limits of Sequences) limn→∞ an = L if and only if for every real number ε > 0 there is a positive integer N such that the numbers aN, aN+1, aN+2, ..., aN+m,... are all within ε of L. The proof is similar to that of the ε, δ condition for limits of functions. The ε, N condition says intuitively that an gets close to L as the integer n gets large. A similar condition can be formulated for limn→∞ an = ∞. THEOREM 3 (ε, δ Condition for Infinite Limits) limn→∞ an = x if and only if for every real number B, there is a positive integer N such that the numbers aN, aN+1, aN+2, ..., aN+m, ... are all greater than B. We conclude this section with another useful criterion for convergence. CAUCHY CONVERGENCE TEST FOR SEQUENCES A sequence <an> converges if and only if (1) aH ≈ aK for all infinite H and K. PROOF First suppose <an> converges, say limn→∞ an = L. Then for all infinite H and K, aH ≈ L ≈ aK. Now assume Equation 1 and let H be infinite. There are three cases to consider.
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Home Infinite Series Sequences Theorems 2 and 3: ε, δ Condition for Limits |