Exponential series – Serlo

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The exponential series has the form . We wil see that he limit indeed exists and is given by Euler's number, which we encountered first within the article Monotony criterion. There, it was defined as the limit of the seqeuences and . In this article, we will show the limit equivalence which is far from obvious. Later, we will investigate a generalized form of the exponential series .

Convergence of the exponential series[Bearbeiten]

At first, we need to show that the exponential series converges at all:

Theorem (convergence of the exponential series)

The series converges.

Proof (convergence of the exponential series)

We need to prove that the partial sums converges. This can be done using the monotony criterion for sequences, where the partial sums form the sequence which we want to investigate.

Monotony is easy to see. Series elements are positive, so

Hence, is monotonously increasing.

Boundedness from above can be shown by comparison to a geometric series with . For partial sums, we have

Hence, is bounded from above by . So the monotony criterion implies convergence.

Limit of the exponential series[Bearbeiten]

Now, we show that the exponential series indeed converges towards Euler's number . This is done using the squeeze theorem by "squeezing" the partial sums between the sequences and . Both bounding sequences converge to , so we get the desired result.

That means, we need to show:

Theorem (limit of the exponential series)

There is .

Proof (limit of the exponential series)

We show
and use the squeeze theorem:

1st inequality: . This is easier to establish than the second one. We need the binomial theorem with .

2nd inequality: . Her, we additionally need the Bernoulli inequality for . In addition, a telescoping sum will appear in the end of the proof.

In addition, we established . Since , the squeeze theorem implies .

Remarks[Bearbeiten]

  • Alternatively, one may show , which also implies .
  • Further, the sequences and define nested intervals , where the real number included in all intervals is exactly .
  • The advantage if the exponential series compared to the sequences defining is that one can achieve much faster convergence. For instance, with 10 elements which is an approximation precise up to 7 digits: . By contrast, the 1000-th sequence element is precise to only 2 digits after the comma.

Outlook: generalized exponential series[Bearbeiten]

As remarked in the introduction, there is a generalization to the exponential series, which reads . This series can be shown to converge for all . Therefore, it serves for a real-valued exponential function . Even complex arguments are possible! However, is a priori not the same as a power . So the computation rules for powers, like , must be shown explicitly.

The series considered within this article is a special case with :