Conditional convergence

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

Definition

More precisely, a series of real numbers

∑

n

=

0

∞

a

n

{\textstyle \sum _{n=0}^{\infty }a_{n}}

is said to converge conditionally if

lim

m

→

∞

∑

n

=

0

m

a

n

{\textstyle \lim _{m\rightarrow \infty }\,\sum _{n=0}^{m}a_{n}}

exists (as a finite real number, i.e. not

∞

{\displaystyle \infty }

or

−

∞

{\displaystyle -\infty }

), but

∑

n

=

0

∞

|

a

n

|

=

∞

.

{\textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=\infty .}

A classic example is the alternating harmonic series given by

1

−

1

2

+

1

3

−

1

4

+

1

5

−

⋯

=

∑

n

=

1

∞

(

−

1

)

n

+

1

n

,

{\displaystyle 1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n},}

which converges to

ln

⁡

(

2

)

{\displaystyle \ln(2)}

, but is not absolutely convergent (see Harmonic series).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem. Agnew's theorem describes rearrangements that preserve convergence for all convergent series.

The Lévy–Steinitz theorem identifies the set of values to which a series of terms in Rn can converge.

Indefinite integrals may also be conditionally convergent. A typical example of a conditionally convergent integral is (see Fresnel integral)

∫

0

∞

sin

⁡

(

x

2

)

d

x

,

{\displaystyle \int _{0}^{\infty }\sin(x^{2})dx,}

where the integrand oscillates between positive and negative values

indefinitely, but enclosing smaller areas each time.

See also

Absolute convergence

Unconditional convergence

References

Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).