Exam 4 Flashcards
Divergence test
if as n approaches infinity isn’t 0 or it is infinity sequence diverges
integral test
take integral of summation from lower bound to upper bound and behavior of the function tells whether regular sequence diverges or not
conditions for integral test
positive function
countinuous function
decreasing function
Comparison test
if a larger version of a function converges the smaller one converges and opposite
P series
1/n^p if p is greater than 1 the series converges
Limit comparison
a.n/b.n
if b.n converges and limit is 0 a.n must also converge
if b.n diverges and limit is inf a.n must also diverge
if neither then test is inconclusive
Remainder for estimating error
Rn < ∫(N-inf)f(x) dx
remainder is less than integral of f(x)
alternating series
if series is decreasing then the limit is 0
Remainder alternating series
|R.n| <= a.n+1
Remainder is less than first term
Root test
p = lim(n appraoches inf) n^ root(a.n)
p is at least 0 but less than 1 converges
p greater than 1 diverges
p = 1 inconclusive
ratio test
r = lim(n approaches inf) a.n+1/a
r less than 1 but at least 0 mean convergence
r>1 means divergence
r =1 mean inconclusive
arithmetic sequence
a.n = a.1 + (n-1)d
geometric sequence
an = ar^(n-1)
harmonic series
1/n diverges
monotone sequence
the function only increases or decreases
monotone convergence theory
if a sequence is bounded and monotone then it converges
telescoping series
the difference of two consecutive terms of a sequence
Geometric series convergence rules
if r is between 1 and -1 series converges
otherwise diverges
Sum rule
two series that converge can be added inside and outside of sigma and have same result
difference rule
two series that converge can be subtracted seperately and maintain same difference
how to add infinite number
a series of partial sums and take limit as n approaches inf
lim(n approach inf) S.n
