When you post a problem, please include the answer choices.
If the infinite sum 1/2^1 + 1/2^2 + 1/2^3 + 1/2^4 +...= 1, what is the value of the infinite sum 1/2^1 + 2/2^2 + 3/2^3 + 4/2^4+....?
A) 1
B) 2
C) 3
D) π (pi)
E) Infinite
Given:
1/2¹ + 1/2² + 1/2³ + 1/2� + ... = 1
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
15/16 + (sum of increasingly small fractions) = 1
(almost 1) +
(sum of increasingly small fractions) = 1.
Implication:
The value in orange is just enough to bring the sum of the lefthand side to 1.
Question:
1/2¹ + 2/2² + 3/2³ + 4/2� + 5/2� + 6/2� + 7/2�...?
=
1/2 + 2/4 +
3/8 + 4/16 +
5/32 + 6/64 + 7/128...
=
1/2 + 1/2 +
3/8 + 2/8 +
5/32 + 3/32 + 7/128...
=
1 +
5/8 +
8/32 + 7/128...
= 1 +
5/8 + 1/4 +
7/128...
= 1 +
(almost 1) +
(sum of increasingly small fractions)
= 1 + 1
= 2.
The correct answer is
B.
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