Sum of the Alternating Harmonic Series (II)

n

3

The striped yellow area is

2n

∑

k=n+1

1

k

, so

2n

∑

k=1

k+1

(-1)

1

k

=

2n

∑

k=1

1

k

-2

n

∑

k=1

1

2k

=

2n

∑

k=n+1

1

k

<ln2

.

The striped blue area is

2n

∑

k=n+1

1

k-1

=

2n-1

∑

k=n

1

k

, so

2n-1

∑

k=1

k+1

(-1)

1

k

=

2n-1

∑

k=1

1

k

-2

n-1

∑

k=1

1

2k

=

2n-1

∑

k=n

1

k

>ln2

.

So

2n

∑

k=1

k+1

(-1)

1

k

<ln2<

2n-1

∑

k=1

k+1

(-1)

1

k

, and therefore

lim

n∞

2n

∑

k=1

k+1

(-1)

1

k

=

∞

∑

k=1

k+1

(-1)

1

k

=ln2

.

Details

Reference: N. Schaumberger, "Mathematics without Words," The College Mathematics Journal, 33(1), 2002 p. 23.

External Links

Alternating Harmonic Series (Wolfram MathWorld)

Permanent Citation

Soledad Mª Sáez Martínez, Félix Martínez de la Rosa

"Sum of the Alternating Harmonic Series (II)"
http://demonstrations.wolfram.com/SumOfTheAlternatingHarmonicSeriesII/
Wolfram Demonstrations Project
Published: March 7, 2011