Question

how do you determine if a series is absolutely ,conditionally convergent or divergent for a problem like this sigma (-1)^n/2^n

Answer 1

The absolute series is a geometric series with ratio 1/2. Hence it is absolute convergent.

Answer 2

how would you test if it is conditionally convergent

Answer 3

since terms go to zero and it alternates it is conditionally convergent.

Answer 4

would you use the latenating test along with another test?

Answer 5

I believe that this is an alternating series, and is convergent. its absolutely convergent if its still convergent using |(-1)^b/2^n| with absolute value operator otherwise its conditionally convergent

Answer 6

if it alternates, which this does because of the
\[(-1)^n\] then all you need for conditional convergence is that the terms go to zero.

Answer 7

okay so you use the alternating test along with another test to see is it is absolutely or conditionally

Answer 8

to show absolute convergence you just check
\[\sum\frac{1}{2^n}\]
which as watchmath said, is a geometric series with \[r=\frac{1}{2}\]

Answer 9

ok i sense it is not clear from your question. absolute convergence is stronger than conditional. it is converges absolutely then it certainly converges conditionally

Answer 10

for an alternating series all you have to check for conditional convergence is that the terms go to zero.

Answer 11

This is an example of series that conditionally convergent
\(\sum (-1)^n\frac{1}{n}\)
The absolute value series is divergent since it is a harmonic series. But the original series is convergent by alternating series test.
So it is always better to check it for the absolute series first.