Harmonic Series Sum Calculator

How it is calculated

Enter the number of terms n, a whole number from 1 to 10,000,000. The tool adds 1 + 1/2 + 1/3 + … + 1/n term by term and also reports the last term 1/n. The sum is shown to 2 decimals. The last term is shown to 2 decimals while it is 0.01 or more, and with about six significant figures once it is smaller, so n = 1000 shows 0.00100000 and n = 10,000,000 shows 0.000000100000. A decimal such as 5.9, zero, a negative number or a value above 10 million returns no result. For large n the sum is close to ln(n) + 0.5772, which grows very slowly: about 5.19 for n = 100, 7.49 for n = 1,000, 16.70 for n = 10,000,000.

Common mistakes

• Assuming the sum settles near a fixed value because the terms shrink. The harmonic series diverges, so the total keeps rising, though extremely slowly.
• Typing a decimal count such as 5.9 and expecting it to be rounded. The number of terms must be a whole number, and anything else shows no result.
• Mixing up the harmonic series with 1/n² or with a geometric series. Those converge, but this tool sums 1/k, not 1/k².
• Entering a huge n to "see divergence". The limit is 10 million terms, where the sum is only 16.70, so the growth is slow rather than visible here.
• Confusing the last term with the sum. The second figure is only 1/n, the size of the final step, not a running total.