Zero sequence criterion

The zero sequence criterion , also known as trivial criterion or divergence criterion , is a convergence criterion in mathematics , according to which a series diverges if the sequence of its summands is not a zero sequence . The zero sequence criterion is a necessary but not a sufficient condition for the convergence of a series.

criteria

The zero sequence criterion is:

If the sequence of the summands of a series does not form a zero sequence, then the series diverges.

So it applies to the summands of a series ${\ displaystyle a_ {i}}$ ${\ displaystyle \ textstyle \ sum \ limits _ {i = 1} ^ {\ infty} a_ {i}}$

{\ displaystyle \ lim _ {i \ to \ infty} a_ {i} \ neq 0}

or if this limit does not exist, the series does not converge. In contrast to other convergence criteria , the zero sequence criterion can only be used to prove that a series is diverging, but not to decide whether it converges. For example, the harmonic series does not converge even though its summands form a zero sequence.

Examples

The series

{\ displaystyle \ sum _ {i = 1} ^ {\ infty} {\ frac {i} {i + 1}} = {\ frac {1} {2}} + {\ frac {2} {3}} + {\ frac {3} {4}} + \ ldots}

diverges, because

{\ displaystyle \ lim _ {i \ to \ infty} {\ frac {i} {i + 1}} = 1 \ neq 0}

.

The alternating series

{\ displaystyle \ sum _ {i = 1} ^ {\ infty} (- 1) ^ {i} = - 1 + 1-1 \ pm \ ldots}

also diverges because the limit value

{\ displaystyle \ lim _ {i \ to \ infty} (- 1) ^ {i}}

does not exist.

proof

The proof of the zero sequence criterion is typically done by counterposition , that is, by reversing the statement

If a series converges, the sequence of the summands forms a zero sequence.

A series converges if the sequence of its partial sums with ${\ displaystyle (s_ {n}) _ {n \ in \ mathbb {N}}}$

{\ displaystyle s_ {n} = \ sum _ {i = 1} ^ {n} a_ {i}}

converges, that is, there is a limit such that ${\ displaystyle s}$

{\ displaystyle \ lim _ {n \ to \ infty} s_ {n} = s}

.

By rearranging the series and using the calculation rules for limit values then applies

{\ displaystyle \ lim _ {n \ to \ infty} a_ {n} = \ lim _ {n \ to \ infty} (s_ {n} -s_ {n-1}) = \ lim _ {n \ to \ infty} s_ {n} - \ lim _ {n \ to \ infty} s_ {n-1} = ss = 0.}

Since the sequence of summands has to form a zero sequence for every convergent series, a series diverges if this is not the case.

Alternative proof using the Cauchy criterion

The trivial criterion can also be proven using the Cauchy criterion . According to this criterion, a series converges if and only if there is a minimum index for all such that is for all . If we put here , it follows: For all there is one , so that the inequality is fulfilled for all . But this is exactly the definition that the sequence is a null sequence. ${\ displaystyle \ epsilon> 0}$ ${\ displaystyle N \ in \ mathbb {N}}$ ${\ displaystyle \ textstyle \ left | \ sum _ {k = m} ^ {n} a_ {k} \ right | <\ epsilon}$ ${\ displaystyle n \ geq m \ geq N}$ ${\ displaystyle n = m}$ ${\ displaystyle \ epsilon> 0}$ ${\ displaystyle N \ in \ mathbb {N}}$ ${\ displaystyle n \ geq N}$ ${\ displaystyle | a_ {n} | <\ epsilon}$ ${\ displaystyle \ left (a_ {n} \ right) _ {n \ in \ mathbb {N}}}$