1. Ch. 10 – Infinite Series 10.4 – Radius of Convergence

2. Convergence Test #1: n th Term Test diverges if. We already knew this! This theorem only gives info on divergence, not on convergence. Ex: Use the nth term test to determine if the following series diverge. DIVERGES! The terms of a n decrease to zero… The results of this test are inconclusive!

3. Convergence Test #2: Geometric Series Test converges if |r|<1 and diverges otherwise. Once again, we already knew this. Ex: Do the following series converge? r > 1, so DIVERGES! r = 1/3 < 1, so CONVERGES!

4. Convergence Test #3: p-series Test converges if p>1 and diverges otherwise. If p = 1, this series is the harmonic series Ex: Use the p-series test (or p-test) to determine if the following series converge. p ≤ 1, so DIVERGES! p = 3/2 >1, so CONVERGES!

5. Convergence Test #4: Ratio Test Let be a series with positive terms, and with. Then… …the series converges if L<1. …the series diverges if L>1. …the test is inconclusive if L=1. Ex: Use the ratio test to determine whether or the series converges: Thus the series converges.

6. Ex: Determine whether the following series converge. – Use p-series test, with p = 3/2… – The p-series test says this series CONVERGES! – The n th term test shows that this series DIVERGES! – Use ratio test! – Since the limit of the ratio test is greater than 1, the series DIVERGES by the ratio test!

7. Convergence Test #5: Direct Comparison Test Let be a series with no negative terms. Then… … converges if there is a convergent series with for all n greater than some integer. …diverges if there is a divergent series of nonnegative terms with for all n greater than some integer. Ex: Prove that converges. Find another function to compare this one to, and find a value of n for which the DC test holds…a function that we know converges or diverges… For, we know Since is a convergent geometric series, the DC test says that the smaller series must also converge.

8. Ex: Determine whether the following series converges. – Use DC test! – Since converges by the p-series test, the DC test says this series CONVERGES!

9. Interval of Convergence Ex: What is the interval of convergence for ? – We finally learn how to find int.s of conv.! – Note that we only find IoC for series with an x in it! To find IoC for power series (like the one above), use the ratio test. Whatever your limit is, set it between -1 and 1 and solve for x. – In this problem, our ratio test limit would be: – Therefore, the IoC would be:

10. Ex: Find the interval of convergence for. – Therefore, the IoC would be -1 < 0 < 1…what does that mean? – Since the limit from the ratio test is a constant between -1 and 1, the IoC is. Ex: Find the interval of convergence for. – Since this limit will only be within (-1, 1) when x = 0, the IoC is x = 0.

11. Ex: Find the interval of convergence and the radius of convergence for – Radius of convergence = ½ the distance between the 2 end values in the IoC – Use ratio test to find IoC first… IoC: RoC =

12. Absolute Convergence If converges, thenconverges absolutely, which is a specific way of saying it converges (for now). Ex: Determine whether converges or diverges. We can’t use ratio test or DC test because some terms may be negative, so find a comparable series with no negative terms by using absolute value bars: Since the series on the right converges to e (think: Maclaurin series!), we can use the direct comparison test to prove the series on the left converges. If converges, then converges.