1. The Harmonic Sequence The Harmonic Sequence is the sequence of reciprocals of the positive integers.

2. The Harmonic Sequence The limit of the Harmonic Sequence, as n approaches infinity, is zero. This means that as n increases, the values in the sequence get closer and closer to zero. 0.000000000001 1,000,000,000,000 0.000001 1,000,000 0.001 1000 0.01 100 H n n

3. You can use a calculator to investigate limits… To examine the Harmonic Sequence, set Then press 2 nd -> TBLSET to decide what values of x (which has replaced n ) to look at. We want to look at large numbers.

4. Not all sequences have limits as n->∞. Those that do are said to converge . If they do not, they are said to diverge . The sequence a n = n diverges, because the values increase without bound. Examine the following sequences. State whether the sequence converges or diverges. If it converges, find the limit: Diverges – the limit does not exist…

5. Properties of Limits

6. How can we put this to use?

7. Now, try it with these:

8. What’s that you say? More?