3.5. Special Sequences
Definition 3.5.7: Exponential Sequence
Proof:
We will use a simple substitution to prove this. Letx/n = 1/u, or equivalently, n = u xThen we have
But the term inside the square brackets is Euler's sequence, which converges to Euler's number e. Hence, the whole expression converges to ex, as required.![]()
In fact, we have used a property relating to functions to make this proof work correctly. What is that property ?
If we did not want to use functions, we could first prove the statement for x being an integer. Then we could expand it to rational numbers, and then, approximating x by rational number, we could prove the final result.