Final exam review sheet

Final exam review sheet - Error bound for a series 0 S-Sn(n...

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= - 11 x (-1,1) ! = e x (- ∞,∞ ) nx2n+12n+1! = sin(x) (- ∞,∞ ) nx2n2n! = cos(x) (- ∞,∞ ) nx2n+1(2n+1) = tan -1 (x) [-1,1] nn = -ln(1-x) [0,2) nx2n22n(n!)2 = J 0 (x) Bessel function (- ∞,∞ ) ddxn=0 cnxn = = ∞ - n 0 ncnxn 1 n=0 cnxn = = ∞ + + n 0 cnn 1xn 1 Error bound for a series: 0 S-S n (n to oo) f(x)dx ; S ≤∫ n S S n + (n to oo) f(x)dx Comparison test for series Alternating Series tests for convergence Suppose a n satisfies: 1. A n >0 2. A n+1 <a n 3. Lim (n->oo) a n = 0 Then (-1) n a n converges Alternating Series Estimation Theorem If S = = ∞(- ) n 1 1 n-1 b n is the sum of an alternating series that satisfies 1. 0 b n+1 b n 2. →∞ limn b n = 0 then R n = S – S n b n+1 Difference Equation p n+1 = kp n (1-p n ) Constant solutions: p n =0 and p n = (k-1)/k Limit Comparison Test Suppose a n and b n are series with positive terms. If →∞( limn a n /b n ) = c, where c is a finite number, then either both series converge or both series diverge. Ratio Test Suppose →∞( + limn an 1an ) = L. If L<1, then the series converges. If L>1, then the series diverges. If L=1, then the
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