Find the sum of the infinite series if there is one. 1/2 +1+2+4...

Expert Answers
hala718 eNotes educator| Certified Educator

S= 1/2 + 1 + 2 + 4 + 8+ ....

S = Sum 2^(n-1)   where n=0,1,2,....

The series is diverge to infinity, then there is no sum to the series. Or the sum is infinity.

However, there is a theorem suggests a sum of ( -1) to the series (1+2+4+8+....). It could be performed as follow:

Let S be the series , so:

s= 1/2 + 1+ 2+ 4+ 8+ 16+....

s = 1/2 + 1 + 2(1+2+4+8...)

s = 1/2 + 1 + 2s

2-2s = 1/2 +1

-s = 3/2

s= -3/2

neela | Student

To find the sum of the series : 1/2+2+4

Solution:

The given sries is a geometric series.  The starting term a =1/2 and nth term a*r^(n-1) and  r = 2.

Therefore sum to n terms  Sn = a(r^n-1)/(r-1) = (1/2) (2^n-1)/(2-1)

= 2^(n-1) -1/2.

Sn = 2^(n-1) -1/2

Sum of infinite terms = Lt Sn = 2^(n-1) -1/2 as n--> infinity is obviously also infinite.