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Show that the harmonic series diverges using: 1) integral test; 2) comparative

Oct 23, 2024

Show that the harmonic series diverges using: 1) integral test; 2) comparative

Solution by Steps

step 1

To show that the harmonic series diverges using the integral test, we consider the integral

\int_1^\infty \frac{1}{x} \, dx

step 2

Evaluating the integral, we find

\int_1^\infty \frac{1}{x} \, dx = \lim_{t \to \infty} \left[ \ln(x) \right]_1^t = \lim_{t \to \infty} (\ln(t) - \ln(1)) = \infty

step 3

Since the integral diverges, by the integral test, the harmonic series

\sum_{n=1}^\infty \frac{1}{n}

also diverges

step 4

For the comparative test, we compare the harmonic series with the series

\sum_{n=1}^\infty \frac{1}{2n}

, which is a constant multiple of the harmonic series and also diverges

step 5

Since

\frac{1}{n} \geq \frac{1}{2n}

for all

n \geq 1

, by the comparison test, the harmonic series diverges

Answer

The harmonic series diverges.

Key Concept

The harmonic series diverges, which can be shown using the integral test and the comparison test.

Explanation

The integral test shows that the integral of

\frac{1}{x}

diverges, leading to the conclusion that the harmonic series diverges. The comparison test further confirms this by comparing it to a known divergent series.