Comparison tests for series

  • Thread starter PFuser1232
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  • #1
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Main Question or Discussion Point

Comparison Test

Consider the two infinite series ##\sum a_n## and ##\sum b_n## with ##a_n,b_n \geq 0## and ##a_n \leq b_n## for all ##n##.
If ##\sum b_n## is convergent so is ##\sum a_n##.
If ##\sum a_n## is divergent so is ##\sum b_n##.

Limit Comparison Test

Consider the two infinite series ##\sum a_n## and ##\sum b_n## with ##a_n \geq 0## and ##b_n > 0## for all ##n##. Define:
$$c = \lim_{n \rightarrow \infty} \frac{a_n}{b_n}$$
If ##c## is positive and finite then either both series converge or both series diverge.

Does it matter if the two series we're comparing have different starting points? I don't think it matters because we can always strip out a finite number of terms from one of the series until both series have the same starting point.

http://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx

But according to this link, "in order to apply this test we need both series to start at the same place."
 

Answers and Replies

  • #2
pwsnafu
Science Advisor
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Does it matter if the two series we're comparing have different starting points? I don't think it matters because we can always strip out a finite number of terms from one of the series until both series have the same starting point.
You are removing a finite number of terms because you care about different starting points.
 
  • #3
Ssnow
Gold Member
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Sometimes it possible to change index in order to have series from a determinate point, as example from ##\sum_{i=1}^{\infty}a_{i}## putting ##i-1=k## we have ## \sum_{k=0}^{\infty}a_{k+1}##...
 
  • #4
33,097
4,796
Does it matter if the two series we're comparing have different starting points? I don't think it matters because we can always strip out a finite number of terms from one of the series until both series have the same starting point.
The limit is as ##n \to \infty##, so the starting points of the two series don't matter.
 

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