Determine whether each series is converging or diverging.

Question

Determine whether each series is converging or diverging. *show steps please*

1. 64, 16, 4, 1, ...
2. 1/4, 1/2, 1, 2, ...
3. 7, -14, 28, -56, ...

Answer 1

1. converges to zero. It's a geometric sequence with common ratio r=1/4
2. diverges approaches infinity. It's a geometric sequence with common ratio r=2
3. diverges oscillates between negative & positive infinity. It's geometric ratio r=-2

If -1 < r < 1, the geometric sequence converges

if not, it diverges

but you asked if the geometric SERIES converges, not the sequence. convergence of the sequence refers to the limit of the value of the last term in the sequence. convergence of the series refers to the limit of the sum of all terms of the sequence.

the Series only converges if -1 < r < 1. Only 1 converges as its r=1/4, -1 < 1/4 < 1

an infinite geometric series = a/(1-r) where a is the 1st term and r = the common ratio

if a/(1-r) is a real number the series converges

so

for 1 a(1-r) = 64/(1-1/4) = 64(4/3) = 256/3. the series converges to 256/3 = 85 1/3

for 2 the sum diverges to positive infinity. the series approaches infinity, never converging to any real number

for 3 the sum oscillates between - and + infinity. It diverges.