The document discusses infinite series and the divergence test. It notes that if the limit of the terms does not equal 0 as n approaches infinity, then the series must diverge. For example, the harmonic series of 1/n diverges even though the limit of the terms is 0. The document then discusses power series and lists learning objectives about finding the general term, writing infinite geometric series in sigma notation, and determining if they converge or diverge based on the common ratio.
4. Divergence TestThe fact that lim𝑘->∞𝑎𝑘=0 does not guarantee that an infinite series converges. However, if lim𝑘->∞𝑎𝑘≠0, the series must diverge! For example, lim𝑘->∞1𝑘=0, but lim𝑘->∞𝑛=1𝑘1𝑛 diverges.
12. Practice: 7,9,11,19,27,33Learning Objectives:Given the first 5 terms of an infinite geometric series, 𝑛=1∞𝑎𝑟𝑛−1Find the general term of the series, 𝑎𝑛 (this 𝑎 and the 𝑎 above are different)Write the series in sigma notation, identifying the initial term (𝑎) and the common ratio (𝑟).Determine whether the series converges or diverges. Is 𝑟<1?If the series converges, find lim𝑛->∞𝑠𝑛=𝑎1−𝑟.
