Flashcards
MATH 101
Cards in this deck (24)
- 1sqrt(a^2-x^2)x=asintheta
dx=acostheta(dtheta) - 21/a^2+x^2x=atantheta
dx=asec^2theta(dtheta) - 3sqrt(x^2-a^2)x=asectheta
dx=asecthetatantheta(dtheta) - 4sqrt(a^2+x^2)x=atantheta
dx=asec^2theta(dtheta) - 5a(x+d)^2+ed=b/2a
e=c-(b^2/4a) - 6Trapezoidaldelta x
first and last terms 1/2, all others 1
start at x_0 - 7Simpson'sdelta x /3
first and last terms 1 (x_0 and x_n)
odd terms 4
even terms 2
n=even - 8Limit TestGiven an integral with infinity as one of the bounds
Replace infinity with R or T
Take the limit as R/T approaches infinity
If a finite limit exists, the integral converges
If the limit DNE or goes to an infinity the integral is divergent - 9Comparison TestLook at dominant terms in the numerator and denominator
Compare to a series with known convergence/divergence, such as a p series - 10Divergence TestTake the limit as n approaches infinity of a_n
if the limit is non-zero the series is divergent
Otherwise the test is inconclusive - 11Integral Testall x must be greater than some N
f(x) must decrease as x increases, and be greater than or equal to zero
Then, using N as a lower bound and infinity as the upper bounds, the series will have the same convergence/divergence of the integral - 12Limit Comparison Test (with integrals)if the integral from a-infinity of g(x) converges, and the limit of f(x)/g(x) as x approaches infinity exists, then the integral from a-infinity of f(x) converges too
if g(x) is divergent, and the same limit exists and is non zero, then f(x) is divergent to0 - 13Limit Comparison Test (with series)If L is the limit as n approaches infinity of a_n/b_n
If L is between zero and inf. a_n has the same div/conv as b_n
If L=0 and b_n is conv., a_n is also conv
If L=inf and b_n is div., a_n is also div
Useless if the limit DNE or b_n isnt divergent, but the limit is infinity - 14Ratio TestLet r be the limit as n approaches infinity of a_n+1/a_n
If r<1 = convergent
If r>1 = divergent
If r=1 or DNE the test is useless
use when x^n or n! - 15Telescoping SeriesS 1-inf of a_n-a_n+1
If the limit as n approaches infinity of a_n = A then, S=a_1-A - 16P-SeriesSum/integral from 1-inf (1/n)^p
If p>1 = converges
If p<(or equal to)1 = diverges
n has to greater than or equal to one
Sum/integral from 0-1
p<1 = converges
p>(or equal to)1 = diverges - 17Geometric SeriesSum from 0-inf where a_n=(a)r^n
S=1/1-r, |r|<1
If N-inf then S=1-r^(N+1)/1-r
*change index if not zero - 18Alternating SeriesSum from 1-inf of (-1)^(n-1)a_n or (-1)^(n)a_n
Converges if a_n>0 for all n, if the successive terms are decreasing, and if the limit as n approaches infinity of a_n is zero - 19arctan(inf)pi/2
- 20Integral of 1/1+x^2arctan(x)
- 21Integral 1/sqrt(1-x^2)arcsin(x)
- 22the sum of x^n1/1-x
|x|<1 - 23Power SeriesA_nx^n = centered at 0
A_n(x-c)^n = centered at c - 24sin(pi+theta) cos(pi+theta)-sin(theta)
-cos(theta)
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- Mathematics Flashcards on MATH 101, created by Charlotte Alexander on 24/04/2023.
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- Charlotte Alexander
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