quartz/content/vault/convergence and divergence of series..md

#math/calculus

$\epsilon$=tiniest little value N = a possible findable number n = index in the sequence. L = a possible findable limit

convergence (for sequences)

for all \epsilon>0 we can find N: n>N ------- |L-a_n|<\epsilon

divergence (for sequences)

for all M we can find N such that (st): n>N ------ $a_n>M$ The sequence constantly grows in a direction. Or, it can oscillate! sin(x) does not converge, so it's divergent.

test for divergence or convergence of series

from sympy import *
x=symbols('x')
series = Sum(1/(6 + exp(-x)), (x, 1, oo))
series.is_convergent()

• test for divergence
• break it into parts
• check if geometric
• telescoping
• MCT monotone_and_bounded? Integral bigger converges? Series converges
• Integral test. If the integral starting from 1 converges, it converges. For the Integral test, the function must be:
  • continuous
  • positive
  • decreasing !
• !More tests for convergence#Direct Comparison Test
• !More tests for convergence#Limit Comparison Test
• !Alternating Series Test
• !Ratio test
• Root Test !Pasted image 20220526191532.png