9.5 Part 1 Ratio and Root Tests Absolute Convergence

You’ve all seen the Harmonic Series converges which ? despite the fact that diverges which only proves that a series might converge. What if we were to alternate the signs of each term? How would we write this using Sigma notation?

This is called an Alternating Series The signs of the terms alternate. Does this series converge?. Notice that each addition/subtraction is partially canceled by the next one. …smaller and smaller… The series converges as do all Alternating Series whose terms go to 0

Alternating Series Test The signs of the terms alternate. This series converges by the Alternating Series Test. Alternating Series Test If and For the series: Then the series converges This is called the Alternating Harmonic Series which as you can see converges…unlike the Harmonic Series

Since each term of a convergent alternating series moves the partial sum a little closer to the limit: Very good! A geometric series. is what kind of a series? What is the sum of this series? Try adding the first four terms of this series: Which is not far from 0.75…but how far?

Since each term of a convergent alternating series moves the partial sum a little closer to the limit: Which is not far from 0.75…but how far? Which means what? Can we agree that

Alternating Series Estimation Theorem leads us to this: Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. Which means that for an alternating series… If k terms were generated then the error would be no larger than term k + 1 Stopped here and where k <  in which is the truncation error then

Alternating Series Estimation Theorem leads us to this: Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. For example, let’s go back to the first four terms of our series: What is the maximum error between this approximation and the actual sum of infinite terms? Answer: Which is bigger than the actual error which we determined to be

Alternating Series Estimation Theorem Since each term of a convergent alternating series moves the partial sum a little closer to the limit: Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. So what does this answer tell us? It simply means that the error will be no larger than Answer: That is why it is commonly referred to as the Error Bound

Alternating Series Estimation Theorem Since each term of a convergent alternating series moves the partial sum a little closer to the limit: Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. This is also a good tool to remember because it is easier than the LaGrange Error Bound…which you’ll find out about soon enough… Muhahahahahahaaa!

Geometric series have a constant ratio between terms Geometric series have a constant ratio between terms. Other series have ratios that are not constant. If the absolute value of the limit of the ratio between consecutive terms is less than one, then the series will converge. This is called the RATIO TEST For the series if then This test is ideal for factorial terms if the series converges. if the series diverges. if the series may or may not converge.

This also works for the nth root of the nth term This also works for the nth root of the nth term. This is called the ROOT TEST(Ex. 57 pg. 508) then For the series if if the series converges. if the series diverges. if the series may or may not converge.

“If a series converges absolutely, then it converges.” Absolute Convergence If converges, then we say converges absolutely. If converges, then converges. If the series formed by taking the absolute value of each term converges, then the original series must also converge. “If a series converges absolutely, then it converges.”

Conditional Convergence If converges but diverges, then we say that converges conditionally If diverges, then diverges. The Alternating Harmonic Series is a perfect example of Conditional Convergence