Direct comparison test

Incalculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (real-valued) terms:^[1]

• If the infinite series ${\displaystyle \sum b_{n}}$  converges and ${\displaystyle 0\leq a_{n}\leq b_{n}}$  for all sufficiently large n (that is, for all ${\displaystyle n>N}$  for some fixed value N), then the infinite series ${\displaystyle \sum a_{n}}$  also converges.
• If the infinite series ${\displaystyle \sum b_{n}}$  diverges and ${\displaystyle 0\leq b_{n}\leq a_{n}}$  for all sufficiently large n, then the infinite series ${\displaystyle \sum a_{n}}$  also diverges.

Note that the series having larger terms is sometimes said to dominate (oreventually dominate) the series with smaller terms.^[2]

Alternatively, the test may be stated in terms of absolute convergence, in which case it also applies to series with complex terms:^[3]

• If the infinite series ${\displaystyle \sum b_{n}}$  is absolutely convergent and ${\displaystyle |a_{n}|\leq |b_{n}|}$  for all sufficiently large n, then the infinite series ${\displaystyle \sum a_{n}}$  is also absolutely convergent.
• If the infinite series ${\displaystyle \sum b_{n}}$  is not absolutely convergent and ${\displaystyle |b_{n}|\leq |a_{n}|}$  for all sufficiently large n, then the infinite series ${\displaystyle \sum a_{n}}$  is also not absolutely convergent.

Note that in this last statement, the series ${\displaystyle \sum a_{n}}$  could still be conditionally convergent; for real-valued series, this could happen if the a_n are not all nonnegative.

The second pair of statements are equivalent to the first in the case of real-valued series because ${\displaystyle \sum c_{n}}$  converges absolutely if and only if ${\displaystyle \sum |c_{n}|}$ , a series with nonnegative terms, converges.

The proofs of all the statements given above are similar. Here is a proof of the third statement.

Let ${\displaystyle \sum a_{n}}$  and ${\displaystyle \sum b_{n}}$  be infinite series such that ${\displaystyle \sum b_{n}}$  converges absolutely (thus ${\displaystyle \sum |b_{n}|}$  converges), and without loss of generality assume that ${\displaystyle |a_{n}|\leq |b_{n}|}$  for all positive integers n. Consider the partial sums

${\displaystyle S_{n}=|a_{1}|+|a_{2}|+\ldots +|a_{n}|,\ T_{n}=|b_{1}|+|b_{2}|+\ldots +|b_{n}|.}$ 

Since ${\displaystyle \sum b_{n}}$  converges absolutely, ${\displaystyle \lim _{n\to \infty }T_{n}=T}$  for some real number T. For all n,

${\displaystyle 0\leq S_{n}=|a_{1}|+|a_{2}|+\ldots +|a_{n}|\leq |a_{1}|+\ldots +|a_{n}|+|b_{n+1}|+\ldots =S_{n}+(T-T_{n})\leq T.}$ 

${\displaystyle S_{n}}$  is a nondecreasing sequence and ${\displaystyle S_{n}+(T-T_{n})}$  is nonincreasing. Given ${\displaystyle m,n>N}$  then both ${\displaystyle S_{n},S_{m}}$  belong to the interval ${\displaystyle [S_{N},S_{N}+(T-T_{N})]}$ , whose length ${\displaystyle T-T_{N}}$  decreases to zero as ${\displaystyle N}$  goes to infinity. This shows that ${\displaystyle (S_{n})_{n=1,2,\ldots }}$  is a Cauchy sequence, and so must converge to a limit. Therefore, ${\displaystyle \sum a_{n}}$  is absolutely convergent.

The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and gon${\displaystyle [a,b)}$  with b either ${\displaystyle +\infty }$  or a real number at which f and g each have a vertical asymptote:^[4]

• If the improper integral ${\displaystyle \int _{a}^{b}g($x)\,dx}   converges and ${\displaystyle 0\leq f($x)\leq g(x)}   for ${\displaystyle a\leq x<b}$ , then the improper integral ${\displaystyle \int _{a}^{b}f($x)\,dx}   also converges with ${\displaystyle \int _{a}^{b}f($x)\,dx\leq \int _{a}^{b}g(x)\,dx.}  
• If the improper integral ${\displaystyle \int _{a}^{b}g($x)\,dx}   diverges and ${\displaystyle 0\leq g($x)\leq f(x)}   for ${\displaystyle a\leq x<b}$ , then the improper integral ${\displaystyle \int _{a}^{b}f($x)\,dx}   also diverges.

Another test for convergence of real-valued series, similar to both the direct comparison test above and the ratio test, is called the ratio comparison test:^[5]

• If the infinite series ${\displaystyle \sum b_{n}}$  converges and ${\displaystyle a_{n}>0}$ , ${\displaystyle b_{n}>0}$ , and ${\displaystyle {\frac {a_{n+1}}{a_{n}}}\leq {\frac {b_{n+1}}{b_{n}}}}$  for all sufficiently large n, then the infinite series ${\displaystyle \sum a_{n}}$  also converges.
• If the infinite series ${\displaystyle \sum b_{n}}$  diverges and ${\displaystyle a_{n}>0}$ , ${\displaystyle b_{n}>0}$ , and ${\displaystyle {\frac {a_{n+1}}{a_{n}}}\geq {\frac {b_{n+1}}{b_{n}}}}$  for all sufficiently large n, then the infinite series ${\displaystyle \sum a_{n}}$  also diverges.

• Convergence tests
• Convergence (mathematics)
• Dominated convergence theorem
• Integral test for convergence
• Limit comparison test
• Monotone convergence theorem

• Ayres, Frank Jr.; Mendelson, Elliott (1999). Schaum's Outline of Calculus (4th ed.). New York: McGraw-Hill. ISBN 0-07-041973-6.
• Buck, R. Creighton (1965). Advanced Calculus (2nd ed.). New York: McGraw-Hill.
• Knopp, Konrad (1956). Infinite Sequences and Series. New York: Dover Publications. § 3.1. ISBN 0-486-60153-6.
• Munem, M. A.; Foulis, D. J. (1984). Calculus with Analytic Geometry (2nd ed.). Worth Publishers. ISBN 0-87901-236-6.
• Silverman, Herb (1975). Complex Variables. Houghton Mifflin Company. ISBN 0-395-18582-3.
• Whittaker, E. T.; Watson, G. N. (1963). A Course in Modern Analysis (4th ed.). Cambridge University Press. § 2.34. ISBN 0-521-58807-3.