- #1
PFuser1232
- 479
- 20
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."
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."