University:
Shippensburg University of Pennsylvania
Course:
MAT 217 | Statistics I
Academic year:

2021
Views:

264

Pages:

2
Author:

Nesbyenvkk

Common Maclaurin Series ∞ 1 1− x x 1 1 + x2  ( −1) k k =0 ∞ k x 2k k=0 xk ∞ 1 + x + x 2 + x 3 + ( −1,1) 1 − x 2 + x 4 − x 6 + ( −1,1) 1+ x +  k! ex k =0 k sin x ( −1) x2k +1  k =0 ( 2 k + 1) ! cos x ( −1) x 2k  ( 2 k )! k =0 ∞ ln (1 + x ) ( −1)k ∞  ∞  x2 k +1 ∞  ( 2 k + 1)! k=0 ∞ cosh x x 2k  ( 2k ) ! k=0 ( −∞, +∞ ) x3 x5 x7 + − + 3! 5! 7! ( −∞, +∞ ) 1− x2 x 4 x6 + − + 2! 4! 6! ( −∞, +∞ ) x− x 2 x3 x 4 + − + 2 3 4 ( −1,1] x− x3 x5 x7 + − + 3 5 7 [ −1,1] x+ x3 x5 x7 + + + 3! 5! 7! ( −∞, +∞ ) 1+ x2 x4 x6 + + + 2! 4! 6! ( −∞, +∞ ) k x 2k +1 2k + 1 ( −1) k =0 sinh x xk k k =1 tan− 1 x +1 x2 x3 x4 + + + 2! 3! 4! x− k ∞ Interval of Convergence m ( m − 1)( m − k + 1) k x , m ≠ 0,1, 2,  ( −1,1) * k! k =1 * Note: The behavior at the endpoints depends on m: For m > 0 , the series converges at both endpoints; for m ≤ −1 , the series diverges at both endpoints; and for −1 < m < 0 , the series converges conditionally at x = 1 and diverges at x = −1 . (1 + x ) m ∞ 1+ Convergence Tests Name Summary Divergence Test If the terms of the sequence don't go to zero, the series diverges. Integral Test The series and the integral do the same thing. p-series Series converges if p > 1. Geometric Series The series converges if the absolute value of the common ratio is less than 1. Direct Comparison If the larger series converges, so does the smaller. If the smaller Test series diverges, so does the larger. Limit Comparison Test If the ratio of the sequences is positive and finite, then both series do the same thing. Ratio Test Find the ratio of two consecutive terms. If the ratio is less than 1, the series converges. If the ratio is greater than 1, the series diverges. Root Test Take the n-th root of the sequence. If the ratio is less than 1, the series converges. If the ratio is greater than 1, the series diverges. Alternating Series If the sequence alternates, the terms in the sequence are decreasing, Test and approaching 0, then the series converges. Where the word sequence is used, it refers to the terms inside the summation.

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