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2011-02-01T00:00:01
<mdoxml version="1.0"><br></br>
It is well known that if $| x | < 1$ then $$1 + x + x^2 + \dots
+ x^n =\frac{1 - x^{n+1}}{1 - x}$$ and hence (taking limits) we
have the sum of the infinite geometric series $$1 + x + x^2 + \dots
+ x^n + \dots = \frac{1}{1 - x}$$ We are now going to obtain a
similar formula for an infinite product, namely $$(1 + x)(1 +
x^2)(1 + x^4)(1 + x^8)\dots(1+x^{2^n})\dots = \frac{1}{1 - x}$$
Evaluate the product $$(1 - x)(1 + x)(1 + x^2)(1 + x^4)(1 + x^8)$$
Show, by induction, that $$(1 + x)(1 + x^2)\dots(1+x^{2^n}) =
\frac{1 - x^{2^{n+1}}}{1 - x}$$ and hence (taking limits) the given
formula for the infinite product follows. <br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<p>This is an infinite product which has close similarities to the infinite geometric series. It is well known that if $| x |< 1$ then $$1 + x + x^2 + \dots + x^{n} = \frac{1 - x^{n+1}}{1 - x}$$ and hence (taking limits) we have the sum of the infinite geometric series $$1 + x + x^2 + \dots + x^n + \dots = \frac{1}{1 - x}$$<br></br>
</p>
<p><span class="editorial">Graeme from Madras College obtained a similar formula for an infinite product.</span></p>
<br></br>
<br></br>
<p>$$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8)\dots(1 + x^{2^n})\dots = \frac{1}{1 - x}$$ The first step is to evaluate the product of a few terms and then to prove a general result. $$\eqalign{ (1 - x)(1 + x)(1 + x^2)(1 + x^4)(1 + x^8) &=& (1 - x^2)(1 + x^2)(1 + x^4)(1 + x^8) \\ \; &=& (1 - x^4)(1 + x^4)(1 + x^8) \\ \; &=& (1 - x^8)(1 + x^8) \\ \; &=& (1 - x^{16})}$$ The
next step is to use the axiom of mathematical induction to prove the following result: $$(1 + x)(1 + x^2)\dots(1 + x^{2^n}) = \frac{1 - x^{2^{n+1}}}{1 - x}$$ For $n = 1$, $$\eqalign{ \mbox{LHS} &=& (1 + x)(1 + x^2) \\ \; &=& 1 + x + x^2 + x^3 \\ \mbox{RHS} &=& \frac{1 - x^{2^{1+1}}}{1-x} \\ \; &=& \frac{1 - x^4}{1 - x} \\ \; &=& \frac{(1 - x)(1 + x + x^2 +
x^3)}{1 - x} \\ \; &=& 1 + x + x^2 + x^3}$$ So the statement is true for $n = 1$.<br></br>
<br></br>
Now assume it is true for $n = k$ where $k$ is an integer. $$(1 + x)(1 + x^2)\dots(1 + x^{2^k}) = \frac{1 - x^{2^{k+1}}}{1 - x}$$ It follows that $$\eqalign{ (1 + x)(1 + x^2)\dots(1 + x^{2^{k+1}}) &=& \frac{1 - x^{2^{k+1}}}{1 - x}(1 + x^{2^{k+1}}) \\ \; &=& \frac{1 - x^{2\times 2^{k+1}}}{1 - x} \\ \; &=& \frac{1 - x^{2^{k+1+1}}}{1 - x}}$$ Hence, by mathematical induction
the statement holds for any positive integer value of $n$. Taking limits, where $| x | < 1$ $$\lim_{n\rightarrow\infty}x^{2^n}=0$$ so the formula for the infinite product follows, namely: $$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8)\dots(1 + x^{2^n})\dots = \frac{1}{1 - x}$$</p></mdoxml>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=253">This</a>is
a nice generalisaton from GPs and a good exercise in proof by
induction. The algebra falls out neatly.<br></br>
<h3><span style="font-weight: bold;">Key question</span></h3>
What is the limit as $x\to \infty$ of $x^n$?<br></br></mdoxml>
<mdoxml version="1.0">The steps to follow are given in the question.<br></br></mdoxml>
<mdoxml version="1.0"><p>Consider these three algebraic identities</p>
<p>$$(1-x)(1+x+x^2+x^3)\equiv (1-x^4)$$</p>
<p>and</p>
<p>$$(1-x)\big((1+x)(1+x^2)(1+x^4) \big)\equiv (1-x^8)$$</p>
<p>and</p>
<p>$$\cos\left(\frac{x}{2}\right)\cos\left(\frac{x}{4}\right)\cos\left(\frac{x}{8}\right)\cos\left(\frac{x}{16}\right)\equiv\frac{\sin(x)}{16\sin\left(\frac{x}{16}\right)}$$</p>
<p>Prove that they are true for any real number $x$ in the first two cases and any real numbers except multiples of $16\pi$ in the third case.</p>
<p> </p>
<p>Use the ideas in your proofs to write down general forms</p>
<p>$$(1-x)(1+x+x^2+x^3+\dots+x^n)\equiv \quad ???$$</p>
<p>and</p>
<p>$$(1-x)\big((1+x)(1+x^2)(1+x^4)\dots(1+x^{n^2}) \big)\equiv \quad ???$$</p>
<p>and</p>
<p>$$\cos\left(\frac{x}{2}\right)\cos\left(\frac{x}{4}\right)\cos\left(\frac{x}{8}\right)\cos\left(\frac{x}{16}\right)\dots \cos\left(\frac{x}{2^n}\right)=\quad ???$$</p>
<p> </p>
<p>In each case it is possible to write down 'infinite $n$' limiting identities. What are these, and for which values of $x$ are they valid?</p></mdoxml>
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Converging Product
In the limit you get the sum of an infinite geometric series. What
about an infinite product (1+x)(1+x^2)(1+x^4)... ?
Pre-Calculus and Calculus
Infinite products
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Long problems
Pre-Calculus and Calculus
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Using, Applying and Reasoning about Mathematics
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Geometric sequence