Question

Find the sum to infinity of the series 108 36 12 4

Answer 1

In any geometric series, we need two pieces of information to find its sum to the

n

-th term, and thus the sum to infinity.

Firstly, we need the first term,

a

. This is obviously

36

in this expression.

Secondly, we need the common ratio,

r

,. We find this by taking

24

36

=

2

3

and confirm that this is the common ratio by taking

16

24

=

2

3

.

Thus,

a

=

36

and

r

=

2

3

. Also note that

|

r

|

=

2

3

<

1

.

We then apply the formula for sum of geometric series to the

n

-th term when

|

r

|

<

1

:

S

n

=

a

(

1

−

r

n

)

1

−

r

=

36

(

1

−

(

2

3

)

n

)

1

−

(

2

3

)

=

108

(

1

−

(

2

3

)

n

)

The sum to infinity,

S

∞

is defined to be the limit of

S

n

as

n

→

∞

.

n

→

∞

,

(

2

3

)

n

→

0

,

S

n

→

108

−

0

=

108

,

S

∞

=

108

.