WOLFRAM|DEMONSTRATIONS PROJECT

Leibniz Criterion for Alternating Series

n

20

zoom

An alternating series

∞

∑

n=1

n

(-1)

a

n

converges if

a

1

≥

a

2

≥…>0

and

lim

n∞

a

n

=0

. Even partial sums

S

2N

=

2N

∑

n=1

n

(-1)

a

n

form an increasing sequence and odd partial sums

S

2N-1

=

2N-1

∑

n=1

n

(-1)

a

n

form a decreasing sequence; their limit is the same.