Question

Evaluate the sum of the series: 1,1/3,1/9 ……. up to infinity

Answer 1

Answer:

Use the formula for the sum of an infinite geometric series to find:

1

+

1

3

+

1

9

+

...

=

3

2

Step-by-step explanation:

The general term of a geometric series is given by the formula:

a

n

=

a

r

n

−

1

where

a

is the initial term and

r

the common ratio.

The sum of an infinite geometric series is given by the formula:

∞

∑

n

=

1

a

r

n

−

1

=

a

1

−

r

when

|

r

|

<

1

. Otherwise the series does not converge.

In our current example

a

=

1

and

r

=

1

3

, so we find:

∞

∑

n

=

1

(

1

⋅

(

1

3

)

n

−

1

)

=

1

1

−

1

3

=

1

2

3

=

3

2

Answer 2

The sum of the infinite series is $S_{\infty}=\frac{3}{2}$.

Step-by-step explanation:

Given : The series is $1,\frac{1}{3},\frac{1}{9},....\infty$

To find : Evaluate the sum of  series ?

Solution :

The series is $1,\frac{1}{3},\frac{1}{9},....\infty$ geometric series.

The first term is a=1.

The common ratio is $r=\frac{\frac{1}{3}}{1}$

The sum of infinite formula is

$S_{\infty}=\frac{a}{1-r}$

$S_{\infty}=\frac{1}{1-\frac{1}{3}}$

$S_{\infty}=\frac{1}{\frac{2}{3}}$

$S_{\infty}=\frac{3}{2}$

Therefore, the sum of the infinite series is $S_{\infty}=\frac{3}{2}$.

#Learn more

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