Harmonic Series

Harmonic Series - Is Divergent to Infinity

• 𝛴_{1≤𝑖≤∞}(1/𝑖) = 1/1 + 1/2 + 1/3 + … = ∞

Proof

Click here to expand...

Let:

• 𝑠_𝑛 = 𝛴_{1≤𝑖≤𝑛}(1/𝑖)

The sequence (𝑠_𝑛)_𝑛∊ℕ is monotonically increasing.

Show that (𝑠_𝑛)_𝑛∊ℕ is not bounded from above.

• 𝛴_{1≤𝑖≤∞}(1/𝑖) = 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + …
• 𝛴_{1≤𝑖≤∞}(1/𝑖) = 1/1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + … # group terms
• 𝛴_{1≤𝑖≤∞}(1/𝑖) > 1/1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + … # replace the terms in each group by the smallest term in the group
• 𝛴_{1≤𝑖≤∞}(1/𝑖) > 1/1 + 1/2 + (1/2) + (1/2) + …
• 𝛴_{1≤𝑖≤∞}(1/𝑖) > ∞ # since there are infinitely many 1/2