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What is the sum of the infinite geometric series?.

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



User JSBach
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2 Answers

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1/2+1/4+1/8+1/16+...=\sum\limits_(n=1)^\infty ((1)/(2))^n = ((1)/(2))/(1-(1)/(2))=1

We can see that consecutive fractions are made from 1/2 to consecutive powers. Because we begin with 1/2, n=1
We will infinitely add fractions , hence Lemniscate sign.

In other words you cannot find a number which needs to be added to the geometric series to get "1", therefore the answer is 1. I remember the teacher explaining it this way :)
User Iwo Kucharski
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8.0k points
5 votes

Answer:

The sum of the given geometric series is, 1

Explanation:

Geometric sequence states that a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio (r).

The sum of the infinite terms of a geometric series is given by:


S_\infty = (a)/(1-r) ......[1] ;where
0<r<1

Given the series:
(1)/(2)+(1)/(4)+(1)/(8)+(1)/(16)+.....

Since, this series is geometric series with constant term(r) =
(1)/(2)

Since,


((1)/(4))/((1)/(2) ) =(2)/(4) = (1)/(2),


((1)/(8))/((1)/(4)) =(4)/(8) = (1)/(2) and so on....

Here, first term(a) =
(1)/(2)

Substitute the values of a and r in [1] we get;


S_\infty = ((1)/(2))/(1-(1)/(2)) where r =
(1)/(2)< 1


S_\infty = ((1)/(2))/((2-1)/(2))

or


S_\infty = ((1)/(2))/((1)/(2))

Simplify:


S_\infty = 1

Therefore, the sum of the infinite geometric series is, 1

User FeuGene
by
7.9k points

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