# Harmonic Series Calculator

The Harmonic series is:

1 + 1/2 + 1/3 + 1/4 + 1/5...

It is a divergent series, which means that the partial sums of this series do not approach a fixed value. Instead, the sum either continually increases or decreases. In this case, the sum increases even though each new term of the series gets
smaller and smaller.

You might say this is because infinitely many numbers are added. However there are sequences whose partial sums approach a fixed value. An example of this is the sequence:

1 + 1/2 + 1/4 + 1/8 + 1/16...

The partial sums of this sequence approaches (but never crosses) 2

Proof of divergence:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8...

1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8...

The first sequence is greater than the second.

The infinite sum of the second sequence is infinity

Therefore, the infinite sum of the first sequence is also infinity.

Enter a number to calculate the sum of the series to that many values

Enter a number to calculate at which number the sum is more than or equal to your number (the opposite of the first calculator)