Approximations for pi of the form a^1/n?

Question

Just for curiosity, how good an approximation can you find for π using an nth-root of an integer (a^1/n), or ( a^m/n).
State your accuracy (no. of decimal places).

It's surprising that you can find small numbers that do a pretty good job:

10^1/2 = 3.16 (1dp accurate)
31^1/3 = 3.1414 (3dps)
97^1/4 = 3.14 (2dps)
306^1/5 = 3.1416 (4 dps)
3020^1/7 = 3.141549 (3 dps)
29809^1/9 = 3.141591 (5 dps)
...

To find such approximations, simply raise π^n for some n, round up/down to nearest integer, then take the nth root.

(This doesn't really have much application, unless you can suggest one where π appears exponentiated, or you really need an approximation to (say) compute the volume of a hyphersphere :) )

Eric, I just wanted answers to the question precisely as phrased.
Here I am asking about non-series-based approximations. Like I said, mainly out of curiosity.

I know all about the well-known series approximations for π (arctangent, Machin-like, Chudnovksy, Ramanujan) as you can see from:
http://answers.yahoo.com/question/index?qid=20070722034309AAGSbv2
and I know they give much better accuracy. The only place we might sensibly apply the a^1/n form is formulae where it is exponentiated, such as the volume of a hypersphere.

I should have said, yes obviously it's an infinite sequence, so you can get any arbitrary accuracy. I was only trying to motivate people to play around with it.

The relationship between n and dp_accuracy(n) is ~ sublinear, but not monotonic, it's a bit noisy; however there are no huge jumps (I didn't suggest there were).

I was not under any circumstances recommending this as a good or fast-converging approximation. It's just a mathematical curiosity (unless someone can suggest applications, other than the hypersphere volume).
The sequence of integers 'a' corresponds to OEIS "A002160 Nearest integer to pi^n"
http://www.research.att.com/~njas/sequences/A002160

Answer 1

There is no limit to how well you can approximate π by some 1/n power of an integer. For example, 5187848314 3196131920 8626152463 0301356268 6760680405 ^ (1/100) will give π accurate to 52 decimal places. The relationship between n (in 1/n) and accuracy of approximation of π seems fairly linear, rarely more than 1 decimal place accuracy from expected. I guess the idea is to find some power of π where it's significantly more accurate than expected, but after trying powers other than 1/n, I haven't really found anything interesting.

Answer 2

For most purposes you don't need to have pi below 3.1416. However, if you need a really precise version of pi, use this equation;

pi/4 = 1-1/3+1/5-1/7+1/9-,+,......

Answer 3

TRY 355/113=3.1415926...THIS SIMPLE FRACTION IS CLOSER THAN MOST EXPRESSIONS OF PI FOR EVERYDAY USE