proof of equivalence of formulas for exp
We present an elementary proof that:
∞∑k=0zkk!=limn→∞(1+zn)n. |
There are of course other proofs, but this one has the advantage that it carries verbatim for the matrix exponential and the operator exponential
.
At the outset, we observe that
∑∞k=0zk/k! converges by the ratio test.
For definiteness, the notation ez below will refer to exactly this series.
Proof.
We expand the right-hand in the straightforward manner:
(1+zn)n | =n∑k=0(nk)(zn)k | ||
=n∑k=0n⋅(n-1)⋯(n-k+1)nkzkk!=n∑k=0π(k,n)zkk!, |
where π(k,n) denotes the coefficient
1(1-1n)⋅(1-2n)⋯(1-k-1n). |
Let |z|≤M.
Given ϵ>0, there is a N∈ℕ such that whenever n≥N, then
∑∞k=n+1Mk/k!<ϵ/2,
since the sum is the tail of the convergent series eM.
Since limn→∞π(k,n)=1 for k, there is also a N′∈ℕ, with N′≥N, so that whenever n≥N′ and 0≤k≤N, then |π(k,n)-1|<ϵ/(2eM). (Note that k is chosen only from a finite set.)
Now, when n≥N′, we have
|n∑k=0π(k,n)zkk!-∞∑k=0zkk!| | =|n∑k=0(π(k,n)-1)zkk!-∞∑k=n+1zkk!| | |||
≤n∑k=0|π(k,n)-1|Mkk!+∞∑k=n+1Mkk! | ||||
=N∑k=0|π(k,n)-1|Mkk!+n∑k=N+1|π(k,n)-1|Mkk!+∞∑k=n+1Mkk! | ||||
<ϵ2eMN∑k=0Mkk!+n∑k=N+1Mkk!+∞∑k=n+1Mkk! | ||||
(In the middle sum, we use the bound |π(k,n)-1|=1-π(k,n)≤1 for all k and n.) | ||||
<ϵ2eM⋅eM+ϵ2=ϵ.∎ |
In fact, we have proved uniform convergence of
over .
Exploiting this fact we can also show:
Proof.
. Given , for large enough , we have
Since , for large enough we can set above.
Since the exponential is continuous11follows from uniform convergence on bounded subsets of either expression for , for large enough we also have . Thus
Title | proof of equivalence of formulas for exp |
---|---|
Canonical name | ProofOfEquivalenceOfFormulasForExp |
Date of creation | 2013-03-22 15:22:52 |
Last modified on | 2013-03-22 15:22:52 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 11 |
Author | stevecheng (10074) |
Entry type | Proof |
Classification | msc 30A99 |
Related topic | ComplexExponentialFunction |
Related topic | ExponentialFunction |
Related topic | MatrixExponential |