Question

How do you show that harmonic series diverges?

Answer 1

We will show that the harmonic series is convergent by proving it by contradiction, if the contradiction of the statement which is that harmonic series converges is proved to be false, and then we can conclude that harmonic series diverges.

Complete step-by-step solution:
We know the harmonic series can be written as:
$\Rightarrow S = \dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + ....$
Now we can split the above given harmonic series as a series which has only odd terms in the denominator and a series which has only even terms in the numerator
Therefore ${S_{even}} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + ....$
And ${S_{odd}} = \dfrac{1}{1} + \dfrac{1}{3} + \dfrac{1}{5} + ....$
And the original series $S$ can be written as: ${S_{even}} + {S_{odd}}$.
Now since we are assuming that the harmonic series converges and each term is positive, we can continue.
$\Rightarrow \dfrac{1}{2}S = \dfrac{1}{2}\left[ {\dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{3} + ...} \right]$
On multiplying we get:
$\Rightarrow \dfrac{1}{2}S = \left[ {\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + ...} \right]$
Which is the ${S_{even}}$ series we made; therefore, we can write it as:
$\Rightarrow \dfrac{1}{2}S = {S_{even}} \to (1)$
Now we know that $S = {S_{even}} + {S_{odd}}$
On rearranging the terms, we get:
$\Rightarrow {S_{odd}} = S - {S_{even}}$
On substituting the value of ${S_{even}}$ from equation $(1)$, we get:
$\Rightarrow S - \dfrac{1}{2}S = \dfrac{1}{2}S$
Which is the value of ${S_{even}}$
Therefore, we can conclude that ${S_{even}} = {S_{odd}} \to (2)$
Now to verify this we will compare both the odd and even series together, on writing them together we get:
$\Rightarrow {S_{odd}} = \dfrac{1}{1} + \dfrac{1}{3} + \dfrac{1}{5} + ....$
$\Rightarrow {S_{even}} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + ....$
Now on comparing term by term in both the series we can see that all the terms in the odd series are greater than the ones in the odd series, we can conclude that:
${S_{odd}} > {S_{even}}$, which is a contradiction because we proved in equation $(2)$ that ${S_{odd}} = {S_{even}}$, therefore the premise that the harmonic series is convergent is false therefore, we can conclude that the harmonic series diverges, rather than converges.

Note: In this proof we use, proof by contradiction, which means that we start with a premise and prove that is false, therefore the contradiction of the premise is true.
The harmonic series can also be represented in the summation form as: $\sum\limits_{n = 0}^\infty {\dfrac{1}{n}}$.