(a) The nth partial sum of a geometric sequence an = arn − 1 is given by Sn =

Question

(a) The nth partial sum of a geometric sequence a_n = ar^n − 1

 is given by S_n = 

 

 .

(b) The series

∞	ark − 1 = a + ar + ar2 + ar3 +
k = 1

is an infinite  ---Select--- arithmetic geometric series. If |r| < 1,

 then this series  ---Select--- converges diverges , and its sum is S = 

 

 .

 If |r| ≥ 1,

 the series  ---Select--- converges diverges .

Answer 1

(a) The nth partial sum of a geometric sequence a_n = ar^n − 1

 is given by S_n = [a₁(1 - rⁿ)] ÷ (1 - r)

(b) The series

∞	ark − 1 = a + ar + ar2 + ar3 +
k = 1

is an infinite  ---Select--- arithmetic geometric series.

If |r| < 1, then this series  ---Select--- converges, and its sum is S = a₁ ÷ (1 - r)

*NOTE: A convergent sequence has a limit — that is, it approaches a real number for the sum. A geometric series converges if and only if -1 < r < 1

For example, If you imagine the common ratio, r = 1/2 and a = 1.

the sum of the geometric series becomes:

S = 1+ 1/2 + 1/4 + 1/8 + 1/16 + ... and so on...

Since |r| < 1, the amount that's added, gets smaller and smaller converging on a real number.

If |r| ≥ 1, the series  ---Select--- diverges .

*NOTE: A divergent sequence doesn't have a limit. — that is, it fails to approach any real number sum.

For example, if you imagine the common ratio, r = 2 and a = 1.

the sum of the geometric series becomes:

S = 1 + 2 + 4 + 8 + 16 + ... and so on..

Since |r| > 1, the amount that's added is larger and larger. The sum fails to approach a real number.