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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,201

Hi;

Here is an interesting numerical means to compute the radius of convergence. It requires that the coefficients of x^n , alternate or are all the same sign.

We need to compute the ordered pairs:

The above generates:

Extrapolate the value in the second column for n = ∞. Looks like we get -.5625.

Here is another one:

This one is a little tougher we must extrapolate to 1 / ∞. So you do a linear fit of the last two points. You get:

The limit of that as x approaches 0 is obviously .2498259

Which is close to the the exact answer of 2.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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