Marvin,

Could you explain the comment by, Steve Finch.

Yes, the sequence of partial sums appears to have
two limit points, which differ by 1 since

lim n^(1/n) = 1  <=== (i can derive this)

-------------
but, what are limit points?
Why do these differ by one?

In my first posting I found odd and even final n
gave different, converging solutions. I believe those are the "two" Mr finch referred to.
The differing by one part? I thought you calculus people had that in hand! I
will find out!...
Ok I got it. It's not calulus it is subtraction.

Look at my convergence table.
Look at the last odd and even entries I have recorded.
.8121...-^-.1879... is about one. It is obvious that when rc is
just about fully coverged the two
will differ by exacty one!!!

The difference between the (n-1)st and nth partial sum is n^(1/n) which goes to 1 as n goes to infinity.

Gregory, thanks.

Hmmm, previous comments seem to have been edited.

Gregory, you are right:)
I edited my own comments that I found to be unecessary, and self evident.

You may find this interesting!


Consider the sequence I named rc:

1^(1/1)-2^(1/2)+3^(1/3)-4^(1/4)...


The sequence of partial sums has two limit points that
differ by one.
Focus on the case where the upper summation is even.

Recording this convergence, I have centered on the powers of two.
I have an accretion of the ascendancy of two up to 30.
 These took over four hours using Mathcad 7 pro. on a Pentium 350MHz.


2^N
(-1)ⁿ*n^(1/n)  =rc
n=1

N                               rc
 1                              .41
 2                              .39
 3                              .33
 4                              .2804
 5                              .244382
 6                              .221224
 7                              .2071156
 8                              .1987905
 9                              .193984
10                              .1912542
11                              .1897242
12                              .1888759
13                              .1884099
14                              .1881559
15                              .1880183
16                              .1879443
17                              .1879046
18                              .1878834
19                              .1878722
20                              .18786625
21                              .18786311
22                              .18786146
23                              .18786059
24                              .187860138
25                              .1878599006789
26                              .1878597767344
27                              .1878597121804
28                              .1878596786128
29                              .1878596611822
30                              .187859652145
rc                              .187859642462
rc^(rc^(rc^...                          .4619214401644

Marvin, a question on your notation.  You used:


2^N
  (-1)n*n(1/n)  =rc
n=1

to indicate a series.  If you are setting (on the bottom of Sigma) n=1 and summing it 2ⁿ times then n = 1,2.  Should it have been written:


  (-1)n*n(1/n)  =rc
n=1


Thank you.

Jeff

To: Jeff Yates


I believe that n is not equal to N. Marv is doing partial sums on a logarithmic scale in his table.

Yes Bob, thank you!

It is now called the MRB Constant.

Intro to the MRB Constant

Let's make this "readable":

0.1 878 596 424 620 671 202 485 179 340 542 732 300 559 030 949 001 387 861 720 046 840 894 772 315 646 602 137
032 966 544 331 074 969 038 423 458 562 580 190 612 313 700 947 592 266 304 389 293 488 961 841 208 373 366 260
816 136 027 381 263 793 734 352 832 125 527 639 621 714 893 217 020 762 820 621 715 167 154 084 126 804 483 635
416 719 985 197 680 252 759 893 899 391 445 798 350 556 135 096 485 210 712 078 444 230 958 681 294 976 885 269
495 642 042 555 864 836 704 410 425 279 524 710 606 660 926 339 748 341 031 157 816 786 416 689 154 600 342 222
588 380 025 455 396 892 947 114 212 218 910 509 832 871 227 730 802 003 644 521 539 053 639 505 533 220 347