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February 19th, 2007, 07:01 AM   #1
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To find the value of an infinite series

friends ,

tel me if it is possible to compute the value of the below series without a calculator/computer.

1 + sqrt(2 + sqrt(3 + sqrt(4 + .........infinite terms ...)

(Note:this series s not diverging .. it is in fact converging)
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February 19th, 2007, 07:49 AM   #2
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I can't see how we would find what this series converged to, since it has no closed form. Any ideas? ( Actually, I think this is a sequence, not a series. )

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February 19th, 2007, 08:51 AM   #3
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hi infinity,
actually i don understand the difference wen u mean as sequence and series ?

anyway, for example say to find the value of the series ,

sqrt(12 + sqrt (12 + sqrt (12 ......to infinite terms )...

this series might seem to diverge ..but not so actually.

LET ,sqrt(12 + sqrt (12 + sqrt (12 ......to infinite terms )... = X

sqrt(12 + X ) = X

therfore, 12 + X = X^2

solvin X = 4 ...... ( u can chek this wit an ordinary calculator for the first few terms itself that it converges to 4)

... but the prob. is that the above method cant b used for the series in our case (but stil usin a calculator, the series (1 + sqrt(2 + sqrt(3 +.....) is seen to converge to 1.75 for first 10 terms) ....

CAN this b got manually by solvin ???
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February 19th, 2007, 09:12 AM   #4
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I'm not sure where you are getting 1.75, because that is less than 1+√2. I took the equation out to about 40 terms and came up with a value of 3.0903...
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February 19th, 2007, 10:22 AM   #5
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Quote:
Originally Posted by roadnottaken
I'm not sure where you are getting 1.75, because that is less than 1+√2. I took the equation out to about 40 terms and came up with a value of 3.0903...

lets keep the series to b of finite terms (for convinience ) say as ,

1 + sqrt(2 + sqrt(3 + sqrt(4 + sqrt(5 + sqrt(6 + sqrt(7 + sqrt(8 + sqrt(9 + sqrt(10) .

comin from reverse.....

>> sqrt 10 = 3.162
>> 9 + 3.162 = 12.162
>>sqrt 12.162 = 3.487
>>8 + 3.487 = 11.487
>>sqrt 11.487 = 3.389
>>7 + 3.389 = 10.389
>>sqrt 10.389 = 3.223
>>6 + 3.223 = 9.223
>>sqrt 9.223 = 3.037
>>5 + 3.037 = 8.037
>>sqrt 8.037 = 2.834
>>4 + 2.834 = 6.834
>>sqrt 6.834 = 2.614
>>3 + 2.614 = 5.614
>> sqrt 5.614 =2.369
>> 2 + 2.369 = 4.369
>>sqrt 4.369 = 2.09
>> 1 + 2.09 = 3.09
>> sqrt 3.09 = 1.758 ...... hope u get it now ....
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February 19th, 2007, 02:41 PM   #6
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A sequence would be something like: n³/e^n from n=1 to n=∞

A series is the sum of all the terms of a sequence, and would be written as:

∑(n=1, n=∞) n³/e^n

I assume that you are trying to find:

Lim n→∞ { n + √(n+1 + √(N+2 + √(N+3 + ..... + √(∞)))) }

Am I correct?
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February 19th, 2007, 02:46 PM   #7
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Quote:
Originally Posted by Infinity
I assume that you are trying to find:

Lim n→∞ { n + √(n+1 + √(N+2 + √(N+3 + ..... + √(∞)))) }

Am I correct?
That looks confusing and not well-defined (the square root of infinity?). Try this:

f(n) = n + sqrt(f(n+1))

Then arun is looking for f(1).
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February 19th, 2007, 03:00 PM   #8
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Well, I just wrote this program to estimate the value of the sequence:

Lim n→∞ { n + √(n+1 + √(N+2 + √(N+3 + ..... + √(∞)))) }

to n = 100,000,000

The answer was:
1.7579327566180045

Code:
import java.util.*;
public class Sequence_Estimation  
{
 public static void main (String[] args) 
 {
  long n=0;
  double answer=0;
  Scanner kbd = new Scanner(System.in);
  System.out.println("Input integer value of 'n' to test sequence to:");
  n = kbd.nextLong();

  for(long i=n; i>0; i--)
  {
   answer = Math.sqrt(answer + i);
  }
  System.out.println("\n" + answer);
 } 
}
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February 19th, 2007, 03:18 PM   #9
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Hey, I just calculated a more accurate answer to n = 10 billion, and guess what? It's exactly the same as the one I reported earlier. Apparently, more decimal places would be needed to show any change.
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February 19th, 2007, 03:56 PM   #10
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The number is called the nested radical constant. Its value is

1.757932756618004532708819638218138527653199922146 8377043
10135500385110232674446757572344554000259452970932 4718478
26956725286405867741108546115435116745974827649802 3843694
89120411842037876481995830644570345768467313417541 5134495
77173273720962022100603227554116598015407552297612 9445796
99112707719478877860007819516309923396999343623052 7753524
96605485188121304121230743966852549640366715265942 2159475
76652412589521440394432605735991324822082490634153 1503978
75302128772604959532494672112007991822456833844067 2864330
74237282346571947808094291349553420592279925860366 1703728
59630816687183328634908728532926587173888717587225 6906069
66741535388517308782986073313679762614334220034550 1474822
19697344628499290204994260780123338419145972718423 7910867
59045639529537528043251120937807502935923611917615 2704264
36487465911939829459953781691083134966345861642367 6784668
18801916873226676954205133566864879409563789163447 6743892
55347895570972640620596122532631802815634393718529 8175824
44581463125494708586493852134993196476027405424112 2516325
98737556657076790516333930301963846032409179377260 1377249
48433124123721498603941391880712274921521093576064 2271839
64712879727605419662075877641516168770731031830438 8844076...
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