MAT136 Final Exam

MAT136 Final Exam
if an = f(n)
∫ {1,∞} f(x) and ∑an both converge or diverge - ANS-Integral test

ar^(n-1), converges to a/1-r if |r| < 1 - ANS-Geometric Series Test

1/n^p, converges if p > 1 - ANS-p-series test

for (-1)^(n+1)bn or (-1)^(n)bn bn > 0. converges if 0 < b_(n+1) <= bn and lim(bn_) = 0 -
ANS-Alternating Series Test

S - S_N <= b_(N+1) - ANS-Remainder in Alternating Series

for p = lim(^nsqrt(|an|)) and p = lim(|a(n+1)/an|) converges if 0 <= p < 1, diverges for p > 1. n/a if
p = 1 - ANS-Ratio/Root Test

Let 0 ≤ an ≤ bn, for all n.
1) If ∑bn converges, then ∑an converges.
Let an >= bn >= 0, for all n.
2) If ∑bn diverges, then ∑an diverges. - ANS-Comparison Test

for positive an, bn
if lim(an/bn) = L not 0 sum(an and bn) both c or d
if = 0, and bn c, an c
if = inf and bn d, an d - ANS-Limit Comparison Test

sum(n=0) f^(n)(a)/n! - ANS-Taylor Series

c at x = a and d for x not a
or c for all x
or c when |x - a| < R, c or d when |x - a| = R - ANS-sum(cnx^n)

cn +- dn c to f+-g
bx^mcn c to bx^mf(x)
c(bx^m)^n c to f(bx^m) - ANS-for cnx^n to f and dnx^n to g