Today we are onto geometric series, which talks about the sum of the first n terms of a geometric sequence. Given a geometric seuqnece with a as its first term, and r as its common ratio, the sum of its first n terms can be written as:

Sn = a(r^n – 1)/(r-1)

We talk about whether a series is convergent (which means the sum is bounded by a constant), or divergent (the series always go bigger or smaller), or it is un-determinant.

When r > 1, the series is divergent;

When r = 1, it is a constant sequence, Sn = a*n, so it is divergent;

When -1 < r < 1, or |r| < 1, the series converges to a/(1-r);

When r = -1, the series jumps between 0 and a, so it is undeterminant;

When r < -1, the series jumps between positive and negative, so it is also undeterminant.

The homework is as follows:

http://www.newtonchineseschool.org/teachers/wangweidong/HomeworkGeometricSeries.pdf