WOLFRAM|DEMONSTRATIONS PROJECT

Leibniz Criterion for Alternating Series

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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.