Limit Comparison Test

• r = lim | a _n / b _n | exists, and 0 < r <

Examples:

• Use the limit convergence test to decide whether the following series converge or diverge. Note that you need to know convergence of the p-series .
  1. Does the series converge or diverge ?
  2. Does the series converge of diverge ?
  3. If r(n) = p(n) / q(n), where p and q are polynomials in n, can you find general criteria for the series p(n) to converge or diverge ?

Since r = lim | a _n / b _n | exists, and r is between 0 and infinity: there exist constants c and C, 0 < c < C < such that for some N > 1 we have:

• c < | a _n / b _n | < C if n > N

• c | b _n | < | a_n | for n > N

Assume converges absolutely. From above we have that

• |a _n | < C | b _n | for n > N