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Sum Term Reciprocals (Posted on 2008-02-22) Difficulty: 3 of 5
A sequence of real numbers {B(m)} is such that:

B(1) = 1, B(2) = 2, and:

B(m+1) = 1 + B(1)*B(2)*..*B(m), whenever m ≥ 2

Evaluate:

Limit (1/B(1) + 1/B(2) + + 1/B(t))
t → ∞

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution -- not proof (spoiler) | Comment 1 of 3

The series converges rather rapidly, to what seems to be 2:

 i      B(i)                   Pi(B(i))                 partial sum
 1               1                         1           1
 2               2                         2           1.500000000000000
 3               3                         6           1.833333333333333
 4               7                        42           1.976190476190476
 5              43                      1806           1.999446290143964
 6            1807                   3263442           1.999999693575066
 7         3263443            10650056950806           1.999999999999906
 8  10650056950807           1.1342371305542 x 10^26   2.000000000000000
 9  1.1342371305542 x 10^26  1.2864938683279 x 10^52   2.000000000000000
10  1.2864938683279 x 10^52  1.6550664732452 x 10^104  2.000000000000000
11  1.6550664732452 x 10^104 2.7392450308603 x 10^208  2.000000000000000

DEFDBL A-Z
total = 1
b = 1
prod = 1
termNo = 1

DO
  termNo = termNo + 1
  b = 1 + prod
  prod = prod * b
  total = total + 1 / b
  PRINT USING "## ############### ############### #.###############"; termNo; b; prod; total
  ct = ct + 1
LOOP

The program quickly crashes on an overflow in prod = prod * b.

 

 


  Posted by Charlie on 2008-02-22 11:35:19
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