The Harmonic Series

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Which of the following best describes the harmonic series?

A finite series formed by summing all positive unit fractions

A series formed by summing all negative unit fractions

A series formed by summing all rational numbers

An infinite series formed by summing all positive unit fractions (correct)

What is the sum of the first n terms of the harmonic series?

Approximately n^3

Approximately n^2

Approximately n!

Approximately ln(n) (correct)

What is the Euler–Mascheroni constant?

A constant equal to 0.577 (correct)

A constant equal to 3.14159

A constant equal to 2.71828

A constant equal to 1.618

Why does the harmonic series not have a finite limit?

Because the series has arbitrarily large values Signup and view all the answers

How was the divergence of the harmonic series proven?

By using the Cauchy condensation test Signup and view all the answers

What is the formula for the sum of the first n terms of the harmonic series?

The sum of the first n terms of the harmonic series is approximately $ln(n) + \gamma$, where $ln$ is the natural logarithm and $\gamma \approx 0.577$ is the Euler–Mascheroni constant. Signup and view all the answers

What does it mean for a series to be divergent?

A divergent series is a series that does not have a finite limit. In other words, the sum of the terms of the series goes to infinity. Signup and view all the answers

Who proved the divergence of the harmonic series?

The divergence of the harmonic series was proven by Nicole Oresme in the 14th century using a precursor to the Cauchy condensation test for the convergence of infinite series. Signup and view all the answers

How can the divergence of the harmonic series be proven using the integral test?

The divergence of the harmonic series can be proven by comparing the sum to an integral, according to the integral test for convergence. Signup and view all the answers