ConvTests - CONVERGENCE TESTS (1) Geometric series: initial...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
CONVERGENCE TESTS (1) Geometric series: initial term = a, common ratio = r a + ar + ar 2 + . . . + = X n = 1 a r n - 1 , X n = 0 a r n . (i) Converges when - 1 < r < 1. (ii) Diverges when r 1 or r ≤ - 1. If convergent a + ar + ar 2 + . . . + = a 1 - r . (2) Divergence Test: lim n →∞ a n 6 = 0 = X n a n diverges (3) Integral test: If f is positive, continuous and decreasing on [1 , ), then Z 1 f ( x ) dx converges ⇐⇒ X n = 1 f ( n ) converges (4) Special Cases: X n = 1 1 n p ± converges p > 1 , diverges p 1 , X n = 1 1 n (ln n ) q ± converges q > 1 , diverges q 1 , (5) Comparison Test: If 0 a n b n , then X n = 1 b n converges = X n = 1 a n converges . If 0 b n a n , then X n = 1 b n diverges = X n = 1 a n diverges . (6) Limit Comparison Test: If 0 a n , b n and lim n →∞ a n b n = L, L 6 = 0 , then X n = 1 b n converges =
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

ConvTests - CONVERGENCE TESTS (1) Geometric series: initial...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online