anonymous
  • anonymous
how do you determine if a series is absolutely ,conditionally convergent or divergent for a problem like this sigma (-1)^n/2^n
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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watchmath
  • watchmath
The absolute series is a geometric series with ratio 1/2. Hence it is absolute convergent.
anonymous
  • anonymous
how would you test if it is conditionally convergent
anonymous
  • anonymous
since terms go to zero and it alternates it is conditionally convergent.

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anonymous
  • anonymous
would you use the latenating test along with another test?
anonymous
  • anonymous
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
anonymous
  • anonymous
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.
anonymous
  • anonymous
okay so you use the alternating test along with another test to see is it is absolutely or conditionally
anonymous
  • anonymous
to show absolute convergence you just check \[\sum\frac{1}{2^n}\] which as watchmath said, is a geometric series with \[r=\frac{1}{2}\]
anonymous
  • anonymous
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
anonymous
  • anonymous
for an alternating series all you have to check for conditional convergence is that the terms go to zero.
watchmath
  • watchmath
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.

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