Determine the sum of an infinite harmonic series.

The sum of an infinite harmonic series diverges to infinity.

An infinite harmonic series is a series of the form:

1 + 1/2 + 1/3 + 1/4 + ...

To determine whether this series converges or diverges, we can use the harmonic series test. This test states that if the terms of a series are of the form 1/n, where n is a positive integer, then the series diverges.

To see why this is the case, we can use the integral test. This test states that if f(x) is a positive, decreasing function, then the series ∑f(n) converges if and only if the integral ∫f(x)dx converges.

In the case of the harmonic series, we can let f(x) = 1/x. This function is positive and decreasing, so we can apply the integral test:

∫1/x dx = ln(x) + C

Since ln(x) diverges as x approaches infinity, the integral diverges, and so does the series. Therefore, the sum of an infinite harmonic series diverges to infinity.

In other words, the sum of the series 1 + 1/2 + 1/3 + 1/4 + ... is infinite. This means that no matter how many terms we add together, the sum will always get larger and larger without bound.