Sum of the Alternating Harmonic Series (II)
Sum of the Alternating Harmonic Series (II)
The striped yellow area is , so =-2=<ln2.
2n
∑
k=n+1
1
k
2n
∑
k=1
k+1
(-1)
1
k
2n
∑
k=1
1
k
n
∑
k=1
1
2k
2n
∑
k=n+1
1
k
The striped blue area is =, so =-2=>ln2.
2n
∑
k=n+1
1
k-1
2n-1
∑
k=n
1
k
2n-1
∑
k=1
k+1
(-1)
1
k
2n-1
∑
k=1
1
k
n-1
∑
k=1
1
2k
2n-1
∑
k=n
1
k
So <ln2<, and therefore
2n
∑
k=1
k+1
(-1)
1
k
2n-1
∑
k=1
k+1
(-1)
1
k
lim
n∞
2n
∑
k=1
k+1
(-1)
1
k
∞
∑
k=1
k+1
(-1)
1
k