Question

Given the series $2+1+\dfrac12+\dfrac14+\cdots$ and the following five statements:

$(1)$ the sum increases without limit.

$(2)$ the sum decreases without limit.

$(3)$ the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small.

$(4)$ the difference between the sum and $4$ can be made less than any positive quantity no matter how small.

$(5)$ the sum approaches a limit.

Of these statements, the correct ones are:

Only $3$ and $4$

Only $5$

Only $2$ and $4$

Only $2,$ $3,$ and $4$

Only $4$ and $5$

Answer: E

Concepts:geometric sequence summation calculus

Difficulty rating: 1670

Small Hint:

Distinguish the terms of the sequence from its partial sums

Big Hint:

A geometric series with first term $2$ and ratio $\frac{1}{2}$ has partial sums approaching $4$

Solution:

The partial sums increase toward $\frac{2}{1-\frac12}=4,$ so they neither increase nor decrease without limit. This makes statements $1$ and $2$ false, while statements $4$ and $5$ are true.

In statement $3,$ “any term” refers to an already selected term, whose distance from zero is fixed and cannot be made smaller; one can instead choose a sufficiently late term below any prescribed positive bound. Thus statement $3$ , as worded, is false.

Therefore only statements $4$ and $5$ are correct, so the correct answer is E.

1950 AMC 12 Problem 39

Answer: E

Difficulty rating: 1670

Small Hint:

Big Hint:

Solution:

Problem 39 in Other Years