Math Problem Solver

Let

{

𝑎

𝑛

}

𝑛

=

1

∞

{a

n

​

}

n=1

∞

​

be a sequence of positive numbers such that:

𝑎

1

≥

𝑎

2

≥

𝑎

3

≥

⋯

≥

𝑎

𝑛

≥

𝑎

𝑛

+

1

≥

⋯

a

1

​

≥a

2

​

≥a

3

​

≥⋯≥a

n

​

≥a

n+1

​

≥⋯ (i.e., the sequence is non-increasing).

lim

⁡

𝑛

→

∞

𝑎

𝑛

=

0.

lim

n→∞

​

a

n

​

=0.

Then the alternating series

∑

𝑛

=

1

∞

(

−

1

)

𝑛

+

1

𝑎

𝑛

∑

n=1

∞

​

(−1)

n+1

a

n

​

is convergent.