Newton is the great virtuoso of infinite power series – what he did with them is astounding, e.g. by age 23 in 1665 found the sine series, cosine series, arc sine series, logarithmic series, binomial series AND the exponential series “which may…be the most important series in mathematics.” Although Indian mathematicians mostly got there hundreds of years before! But no-one in Europe knew that until much later.
The simplest power series is:
Such infinite series are easy to construct, but still I find it kind of mysterious that they add to the fraction on the other side. Well, it’s due to the telescoping.
Then take the denominator over to the other side and start writing in terms to make the RHS add to 1. i.e.
and multiply this out on another line:
Now there’s a to get rid of, so write in a
and the s cancel. Next an :
and the s cancel.
Not hard to see that this will go on forever in exactly the same way.
And we are left with
IF as ! i.e. the series converges. For , that evidently won’t happen.
This is shown visually with the 2 similar triangles in this picture: ST/PS = PQ/QR
I came across this picture (from Roger Nelsen’s book Proofs Without Words) after exploring power series, generating functions, continued fractions etc for a while – it came as a revelation to me. I hadn’t imagined there could be a simple drawing showing how it works! Why hadn’t I seen it before?! There are too many maths books without pictures! Although not so many these days; people realise the huge value of a good diagram.
This one shows the more general case of :
All kinds of interesting results pop out when you plug values in, e.g if ,
This pic from Proofs Without Words II :
I think must be , otherwise the series diverges to infinity, producing some crazy equation. (Although Euler didn’t mind.)
Plugging for into produces:
Multiplying both sides of by gets:
Replacing by in , we get:
Replacing by in gets:
Multiplying both sides of by , we get:
Replacing with in produces:
(Mercator, Brouncker 1668)
Plugging in to this gives:
Reciprocal of a power series
If , there is a power series such that
with coefficients determined recursively