WEBVTT

1
00:00:00.000 --> 00:00:05.341
Here is the problem: Σ 4 (0.5) ^ n, n equals 0 to ∞.

2
00:00:05.791 --> 00:00:13.245
Decide whether Σ 4 (0.5) to the power n (n from 0 to ∞) converges or diverges.

3
00:00:14.045 --> 00:00:26.038
Write the series. Σ a(n) for n equals 0, 1, 2, … with a(n) equals 4 (0.5) to the power n. First terms: 4, 2, 1, 0.5, ….

4
00:00:26.488 --> 00:00:43.415
Recognise a geometric series. Each term is the previous one times r equals 1 over 2 (checked for the first 40 terms), with first term a equals 4. Since |r| less than 1 the series converges, and its sum is a/(1 minus r) equals 8.

5
00:00:44.215 --> 00:00:50.566
So the answer is The series converges (geometric series (|r| less than 1)); sum equals 8.

6
00:00:51.016 --> 00:01:07.375
A series converges when its partial sums approach a finite number. The terms must tend to 0 for that to happen, but that alone is not enough (the harmonic series diverges), so a comparison, ratio, root, alternating or integral test is used to decide.

7
00:01:08.175 --> 00:01:20.609
Partial sums (supporting evidence, not a proof): S₁₀ equals 7.9921875, S₁₀₀ equals 8, S₁₀₀₀ equals 8, S₁₀₀₀₀ equals 8.