Principle of Mathematical Induction/Warning/Example 2

Example of Incorrect Use of Principle of Mathematical Induction
We are to prove that:
 * $\dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 3} + \dotsb + \dfrac 1 {\paren {n - 1} \times n} = \dfrac 3 2 - \dfrac 1 n$

For $n = 1$ we have:
 * $\dfrac 3 2 - \dfrac 1 n = \dfrac 1 2 = \dfrac 1 {1 \times 2}$

Assuming true for $k$, we have:

But clearly this is wrong, because for $n = 6$:


 * $\dfrac 1 2 + \dfrac 1 6 + \dfrac 1 {12} + \dfrac 1 {30} = \dfrac 5 6$

on the, but:
 * $\dfrac 3 2 - \dfrac 1 6 = \dfrac 4 3$

on the.

Refutation
The supposed sequence of terms on the starts at $n = 2$.

It can be seen that it is meaningless (or has no terms for $n = 1$.

Hence in the first statement:
 * $\dfrac 3 2 - \dfrac 1 n = \dfrac 1 2 = \dfrac 1 {1 \times 2}$

the term on the is the term for $n = 2$, making this equation invalid.

The correct result is Sum of Sequence of Products of Consecutive Reciprocals: