cheatsheet - IMPROPER INTEGRALS p if p > 1 converges 1 dx/x...

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IMPROPER INTEGRALS 1 dx/x p if p > 1 converges if p ≤1 diverges Direct Comparison: g(x) ≥ f(x) if ∫g(x)dx C, so does ∫f(x)dx if ∫f(x)dx D, so does ∫g(x)dx Limit Comparison: If lim g(x) = L then a f(x)dx and a g(x)dx x f(x) both C or D INFINITE SERIES 1) Check if lim a n = 0 ; if NO, series is D n 2) Geometric ? Is it of the form: ∑ ar (n-1) = a/(1-r) n=1 if |r| < 1, then converges to (1-r) if |r| ≥ 1, then diverges 3) Telescoping ? Is if of the form: ∑ (a n – a n+1 ) n=1 if a n 0, then sum = a 1 -try rewriting with partial fractions 4) Is it of the form ∑ 1/n p n=1 if p > 1 C; if p ≤ 1 D 5) Direct Comparison Test : If a n > 0 for all n AND a n ≤ c n and ∑ c n is C, n=1 then ∑ a n is C n=1 if a n ≥ d n and ∑ d n is D , then ∑ a n is D n=1 n=1 6) Limit Comparison Test : (put what you are comparing w/ on the bottom ) If lim a n /b n = c > 0, then ∑a n & ∑b n both C or D n = 0 then C of ∑b n guarantees C of ∑a n = ∞ then D of ∑b n guarantees D of ∑a n 7) Ratio Test : (good w/ n!’s) If lim a n+1 /a n <1, then C n >1, then D =1, inconclusive 9) Alternating Series Theorem (check absolute convergence first, then use i) ii) iii) to check for conditional convergence) ∑(-1) n a n converges if: n=1 i) a n > 0 for n>N ii) a n+1 ≤ a n for n>N iii) lim a n = 0 n -interval of C is when the end points are checked (if converges, then is also equal to the endpoint) -absolute interval of C is the interval given by the ratio test used to find the limit -radius is what the limit goes to -conditionally convergent where x makes CC If ∑ a n is C, then series is absolutely convergent If ∑ (-1) n+1 a n is C but ∑a n is D then ∑ (-1) n+1 a n is conditionally convergent . 10) Power Series : power series about b: ∑C n (x-b) n 1)Ratio Test it! n=0 2) get radius of convergence 3)Test end points (plug x’s back into original series & see if series converges with a test): if C, then (=) in interval of convergence and converges conditionally at that x interval of absolute conv. Stays same except if end point x causes absolute convergence (in alt. series) 11) Taylor Series : for f(x) about x=a is ∑ f (n) (a) (x-a) n n=0 n!
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