Bounding Partial Sums of the Harmonic Series

n

1

n

∑

i = 1

1

i

≈ 1.00000000000 ≥ 1 +

1

2

× 0 = 1.

1 ≥ 1

To get a lower bound on the

th

n

partial sum of the harmonic series

n

Σ

i=1

1

i

, replace the sum of the

k

2

terms

1

k

2

+1

+

1

k

2

+2

+…+

1

k

2

+

k

2

by

1

k+1

2

+

1

k+1

2

+…+

1

k+1

2

=

k

2

k+1

2

=

1

2

for each

k

,

0≤k<

log

2

n

. This shows that the

th

n

partial sum of the harmonic series is bounded below by

1+

1

2

⌊

log

2

n⌋

. Move the slider to see the partial sums and this lower bound. Terms of the same color sum to

1

2

.