[Math] Induction Proof for a series expansion of a function

calculusinductionpower seriestaylor expansion

I have done induction proofs of many different types, but trying to prove by induction that a derivative from the Taylor series expansion of a function has me stumped in terms of how to get the final step to work.

In most induction proofs we start with a base case of $n=1$, then assume $N=n$ works, then apply
$N+1$, and usually make use of the assumption to get a statement that is TRUE to prove it.

Here is the function:
$$f(x)= \ln (1+x).$$
And here is the equation we wish to apply Induction to:
$$f^{(n)}(x)= (-1)^{n-1}\frac{(n-1)!}{(1+x)^{n}}.$$

The approach I took was to apply $N+1$ where ever I see $n$, and the other thing that I did was
to just differentiate the equation which was straight forward, and then both were the same.
BUT i don't feel that this was a good proof because in Induction you usually make use of
step 2 the equation itself.

Hope someone can guide me or explain to me how one goes about doing an induction proof that
has derivatives involved of this sort.

Best Answer

The way you've described induction is a bit muddled, and I blame the fact that proofs using induction are often written very informally.

The conceptually simplest form of the principle of mathematical induction is this:

Let $S$ be a set of natural numbers with the following two properties:

  1. $0\in S$

  2. For all $n\in \Bbb N$, if $n\in S$ then $n+1 \in S$.

Then $S = \Bbb N$.

This principle can be recast in logical terms (as a "schema" as follows):

Let $P$ be a propositional function in $\Bbb N$. That is, for each $n\in \Bbb N$, let $P(n)$ be a statement.

Suppose that the following hold:

  1. $P(0)$ is true.

  2. For each $n\in \Bbb N$, if $P(n)$ is true then $P(n+1)$ is true.

Then we can conclude that for each $n\in \Bbb N$, $P(n)$ is true.

In fact, it's easy to see that either of these approaches can be extended to allow any initial value, not just $0$, at the cost of having the proposition hold (or the number be an element of the set) only for numbers greater than some value (in your case $1$).

Using one of these somewhat more formal approaches to induction should help you keep a clear idea in your mind of how you need to approach the proof.

Let $P(n)$ be the statement that $$f^{(n)}(x) = (-1)^{n-1}\frac{(n-1)!}{(1+x)^n}$$. Prove first that $P(1)$ is true. Then prove that for each $n\in\Bbb N$, if $P(n)$ is true then so is $P(n+1)$.

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