Example: Investigate the convergence properties of the following series:

  1. What is the actual limit of the sum ?
  2. What is the actual limit of the sum ?
  3. Does the sum converge ?
The first series seems to be a geometric series with a = 1/2. However, the index n starts at n = 1, whereas for the geometric series it starts at n = 0. While that does not influence the convergence behavior, it does change the actual limit of the series. In fact, the we have that: by the geometric series test, so that The second series is again similar to the geometric series, except for the index, which is supposed to start at 0. This does not influence convergence (or divergence) but it does change the actual value of the series. While we have for our series: which is the answer to the above infinite series.

For the last series we will use the limit comparison test, together with the geometric series test. First note that

Therefore, by the limit comparison test, the series and have the same convergence behavior. But by the geometric series test, the second series converges, so that by the limit comparison test the first one also converges.

Note that we have established convergence of the series, but we do not know the actual limit. In fact, that limit is very difficult to determine.

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