Chapter 11 Flashcards
P series
form: sum 1/n^p
Conditions: if p>1 convergent
if p less than or equal to 1 divergent
Geometric series
form: sum ar^n
conditions: if absolute value of r <1 convergent
if absolute value of r greater than or equal to 1 divergent
equation for sum of geometric sequence
Sn= a1/1-r
Comparison Test
form: similar to p-series of geometric
conditions: terms positive in both series
if sum bn is convergent and an is less than of equal to bn for all n then sum an is convergent (AND VICE VERSA)
Limit Comparison Test
form: similar to p-series of geometric
condiitions: sum an and sum bn both have positive terms. If lim as n approaches infinity an/bn=c, c is finite and positive
*Either both series converge or both series diverge
Divergence Test
Form: lim n–> infinity an DOES NOT EQUAL 0
conditions: Iim does not equal 0
lim DNE
–> divergent
IF IT DOESNT WORK, use another test.
Alternating Series Test
Form: Sum (-1)^n-1 times bn
conditions: bn+1 less than or equal to bn for all n
lim n–>infinity bn=0
–> convergent
Ratio Test
Form: Involving factorials or products (including constants raised to the nth power)
Conditions: if lim n–>infinity absolute value of (an+1/an) = L < 1 absolutely convergent
>1 divergent
=1 inconclusive
Root Test
form: (an)^n
Conditions: lim n–> infinity n root of absolute value of (an)= L < 1 absolutely converges
> 1 or = infinity divergent
Conditional convergence: converges but absolute value diverges
Integral Test
Form: if integral is easily evaluated
Conditions: For an=f(n), f is continuous, positive, and decreasing
If Iim t–>infinity of integral __ to t f(x) dx is convergent then sum an is convergent and VICE VERSA
difference between sequence and series
sequence: a LIST
series: a SUM
monotonic sequence
always increasing or decreasing (not alternating)
