Calc II Exam 2

Definitions of series tests

Geometric Series Test

Geometric Series Test

if r >= 1 - the series diverges

if r < 1 - the series = c/(1-r) and converges

Integral Test

if f(x) is positive and decreasing - series converges if integral converges

if f(x) is positive and decreasing - series diverges if integral diverges

Divergence Test

if lim a(n) != 0 - series diverges

if lim a(n) = 0 - series converges

Direct Comparison Test

if a(n) <= b(n) and series b(n) converges - series a(n) converges

if a(n) <= b(n) and series b(n) diverges- series a(n) diverges

Limit Comparison Test

if a(n) and b(n) are positive - L = lim a(n)/b(n)

if not L>= 0 or L = inf - if a(n) converges then b(n) converges

if not L>= 0 or L = inf - if a(n) diverges then b(n) diverges

if L != 0 or inf - use another test

Alternating Series Test

for series (-1)^n*b(n)

if b(n) > 0 and b(n+1) > b(n) and lim b(n) = 0

then - series (-1)^n*b(n) converges

Ratio Test

L = lim |a(n+1)|/|a(n)|

if L < 1 - series a(n) converges

if L > 1 or lim = inf - series a(n) diverges

if L = 1 - try another test

Root Test

L = lim nsqrt(|a(n)|)

if L < 1 - series a(n) converges

if L > 1 or lim = inf - series a(n) diverges

if L = 1 - try another test

P - Test

If p > 1, then the series converges,

If p ≤ 1, then the series diverges.