13.2Series

Sequence 2, 4, 6, …, 2n, … A sequence is a list. Related finite series Related infinite series … + 2n + … Series is the list as a sum. Every finite series represents a partial sum of a sequence. Partial sum is the total of the terms of a sequence up to a given term.

Example If, what is the sum of the finite series formed by adding the first five terms of the sequence ?

Find for each of the following infinite series. If the sequence were to continue indefinitely, does it seem to have a limit or not? If yes, predict the limit. # … # … # (0.5)^2+(0.5)^3+(0.5)^4+…

Sigma notation is useful for describing series. If the graph of a sequence of partial sums does not approach a horizontal asymptote, then we say that the infinite series does not have a sum.

Summary An infinite sequence can have a limit. This is the value that the sequence’s terms approach as the index goes to infinity. An infinite series can converge or diverge. If the series converges, it has a specific sum; otherwise it does not. These two concepts are related. Associated with any infinite series is an infinite sequence of partial sums. If the infinite sequence of partial sums has a limit, the series is said to converge, with a sum equal to the limit. If the infinite sequence of partial sums does not have a limit, the series to said to diverge and has no sum.

Example Evaluate Example Evaluate

Example Evaluate Example Evaluate