10.3 Convergence of Series with Positive Terms Do Now Evaluate

HW Review

Positive Series We consider a positive series one where every term is greater than 0. If S is a positive series, then either – The partial sums are bounded above, so S converges – The partial sums are not bounded above, so S diverges

Integral Test Let, where f(x) is positive, decreasing, and continuous for x >=1 If converges, then converges Ifdiverges, thendiverges

Ex Determine whether the harmonic series converges or diverges

Ex Determine whether the series converges or diverges

P-Series The infinite series converges if p > 1 and diverges otherwise The integral test is used to prove this theorem

Comparison Test Assume that there exists M > 0 such that for n >= M. Then: If converges, thenconverges Ifdiverges, then diverges

Ex Doesconverge?

Ex Doesconverge?

Limit Comparison Test Let A and B be positive sequences. Assume that the following limit exists: If L > 0, thenconverges if and only if converges If L = infinity and converges, then converges If L = 0 andconverges, thenconverges

Ex Show thatconverges

Closure Determine whether the series converges or diverges HW: p.566 #