Exponential Sequence

Converges to the exponential function e^x = exp(x) for any real number x.

Proof:

We will use a simple substitution to prove this. Let

x/n = 1/u, or equivalently, n = u x

Then 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 e^x, 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.

Specials |

Next Sequence |

Prev. Section |

Glossary |

Map

(bgw)