So we find that pi/4 is inside some interval bounded from below by the 'even' subtotals and from below by the 'odd' subtotals. We technically would need to prove that the second sum converges, but I'll skip that part for clarity. Where (1/(2n+2k+1) - 1/(2n+2k+3)) is strictly positive, so the total error is positive as well. We now look at what this is when n is even and when n is odd. If we add the first n terms then the total error is everything from the n th term onwards. There's actually a rather simple reason that they are all off by about 1/n after the n th term. I should point out that they are not, themselves, alternating series, but the idea is similar, and it allows you to see how growth rates of terms relate to convergence rates of series. Or, check out this Ramanujanian monstrosity which converges very rapidly: With those series, the magnitudes of the terms get much smaller far more quickly. You should try using other series, like summing 1/n 2 to get pi 2 /6.
The magnitudes of the alternating terms grow like 1/(2n), which is indeed quite slowly.
#Calculate pi to the 7th series
After adding 10 n terms of your series, the next term is pretty much exactly 10 -n, hence the decimal accuracy you have observed.įollow up edit: You noted that this series is "slowly convergent", and you should be discovering why this is the case. The main idea is that after adding N terms, you are at a distance away from the ultimate sum that is at most the next term in your sum. This is because you have an alternating series! Read up about it here.
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