1950 AHSME Problems/Problem 39

Problem

Given the series $2\plus{}1\plus{}\frac {1}{2}\plus{}\frac {1}{4}\plus{}...$ (Error compiling LaTeX. ! Undefined control sequence.) 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 statments, the correct ones are:

$\textbf{(A)}\ \text{Only }3 \text{ and }4\qquad \textbf{(B)}\ \text{Only }5 \qquad \textbf{(C)}\ \text{Only }2\text{ and }4 \qquad \textbf{(D)}\ \text{Only }2,3\text{ and }4 \qquad \textbf{(E)}\ \text{Only }4\text{ and }5$

Solution

This series is a geometric series with common ratio $\frac{1}{2}$. Using the well-known formula for the sum of an infinite geometric series, we obtain that this series has a value of $2\cdot \frac{1}{1-\frac{1}{2}}=4$. It immediately follows that statements 1 and 2 are false while statements 4 and 5 are true. 3 is false because we cannot make any term of the sequence approach 0, though we can choose some term that is less than any given positive quantity. The correct answer is therefore $\boxed{\textbf{(E)}\ \text{Only }4\text{ and }5}$.

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