(3/2)-(5/6)+(7/18)-(9/54) +........infinity
Answers
Answer:
Step-by-step explanation:
Answer:
15
16
Explanation:
This is not a geometric series. It has no common ratio, but the intention seems to be that the general term is:
a
n
=
(
−
1
)
n
−
1
(
2
n
+
1
)
(
2
⋅
3
n
−
1
)
If we sum pairs of terms, then the resulting series has all positive terms, with the following expression for the general term:
b
n
=
(
4
n
−
2
)
(
3
⋅
9
n
−
1
)
The numerators are in arithmetic progression with initial term
2
and common difference
4
.
We can split this series into a series of series, hence into a product of a geometric series and a geometric series with constant offset.
Note first that
∞
∑
k
=
0
1
9
k
=
9
8
So we find:
3
2
−
5
6
+
7
18
−
9
54
+
11
162
−
13
486
+
15
1458
−
17
4374
+
...
=
2
3
+
6
27
+
10
243
+
14
2187
+
...
=
1
3
(
(
2
+
2
9
+
2
9
2
+
2
9
3
+
...
)
+
(
4
9
+
4
9
2
+
4
9
3
+
...
)
+
(
4
9
2
+
4
9
3
+
4
9
4
+
...
)
+
(
4
9
3
+
4
9
4
+
4
9
5
+
...
)
+
...
)
=
1
3
(
1
+
1
9
+
1
9
2
+
1
9
3
+
...
)
(
2
+
4
9
+
4
9
2
+
4
9
3
+
4
9
4
+
...
)
=
4
3
(
∞
∑
k
=
0
1
9
k
)
(
−
1
2
+
∞
∑
k
=
0
1
9
k
)
=
4
3
⋅
9
8
⋅
9
−
4
8
=
4
3
⋅
9
8
⋅
5
8
=
15
16