Unit 2 Convergence Tests

Geometric series

a/(1-r)
Convergence if |r| < 1

Diveregence if |r| >= 1

nth Term Test

a_n
Divergence: if limit as n approaches infinity is NOT equal to 0

Inconclusive: if limit as n aproaches infinity is equal to 0

Cannot prove convergence

Integral test

• Do integral of inner function from N value to infinity

• If received a real number then CONVERGENCE

• Otherwise divergence

P-Series Test

Sumation of n=N to infinity is 1/(n^P)

• Converges if P > 1

• Diverges if 0 < P <= 1

• Opposite of Geometric test basically

Alternating Series Test

Sumation of something of (-1)^n * f(x)

• Convergence: sequence must decrease and lim as n aproaches 0 of f(x) = 0

• Cannot prove divergence, do another test

Absolute/conditional convergence:

• Use direct/limit comparison and if that converges as well as the orignal absolute convergence

• If just orignal converges then conditional convergence

• If neither then divergence

Direct Comparison

Pick a function thats the more simplisitc version of the given sumation

Limit Comparsion

Find simpler function and divide it by the original function, take the lim of that

• Convergence: 0<x<infinity (positive and finite value)

• if (a_n)/(b_n) = 0 sumation of a_n converge if b_n converge

• if (a_n)/(b_n) = infinity sumation of a_n diverges if b_n diverge

Ratio Test

take absolute value lim of increase the nth term by one and divide by original function

• Converges if that whole thing is <1

• Diverges if >1 or =infinity

• Same as geometric test basically

Root Test

Take the nth root of the absolute value of a_n

• Converges if that is <1

• Diverges for anything else

• Inconclusive if that is = 1, try smthn else