Prove that the sum of the infinite series 1+1-1+1-1+... is either 1 or 0.
<edited:>
the sum can be 1/2 too..
in fact, the sum is formally defined to be 1/2.
So...go ahead and prove it!
11Take the sum uptil 'n' terms, ans comes out from whether n is odd or even.
1if sum is extend till n 4 +1...den
+1 +1 + 1.....n + -1 -1 -1....(.n-1)
= n - (n-1)
=1
..
if sum of -1 is extendd 2 n
den also
-1 -1 -1.......n + +1 +1...(n-1)
=-n+n-1
= -1
so sum will b alwys +/- 1.
62What shreyan is probably hinting is to write this as S
then
S=1-S
thus, S=1/2
BTW if i remember correctly i had put a QOD on this funda... I could not mine it out! :(
11-1=0
1-1+1-1+1-1+.......
n(1-1)=0 if n is even
n(1-1)+1=1 if n is odd
is s=1/2 physically possible
1@jagaran
S=1/2, is not possible, though the limiting value of S as the number of terms tends to ∞ is 1/2 if U solve it by the above methodology as told by Nishant sir
Though I particularly feel that the limit does not exist. But then if consider it as a series, then the answer is 1/2
The answer can also be obtained by this method.......
The given sequence is a GP with first term 1 and common ratio -1, which gives the sum as..
S=a/1-r =1/(1-(-1))
=1/2
[339]
1@kalyan how can you use formula for gp series here mod(common ratio) is not less than 1 .it is an infinitely oscilating series
62@Jagaran.. what you are saying is absoulutely true..
But this is an infinite series.. so waht will the sum be?
62@kalyan... your conclusion is wrong as given/ shown by Jagaran...
His post is correct
@Jagaran. (But when I am talking to you, I am arguing in the other direction ;)
